A User’s Guide to Algebraic Topology [387] 9780792342939

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A Users' Guide To Algebraic Topology C. T. J. Dodson

University of Manchester Institute of Science and Technology Phillip E. Parker

Wichita State University

Contents Preface Introduction and Overview 1 Basics of Extension and Lifting Problems 1.1 1.2 1.3 1.4 1.5

Existence problems : : : : : : : : : : : Retractions : : : : : : : : : : : : : : : Separation : : : : : : : : : : : : : : : : Transcription of problems by functors The shape of things to come : : : : : :

2 Up to Homotopy is Good Enough 2.1 2.2 2.3 2.4 2.5 2.6

Introducing homotopy : : : : Fibrations and co brations : Commuting up to homotopy : Homotopy groups : : : : : : : The fundamental group : : : First applications : : : : : : :

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3.1 Introduction : : : : : : : : : : : : : : : : : : : : : 3.1.1 Exact sequences : : : : : : : : : : : : : : 3.2 Relative homotopy : : : : : : : : : : : : : : : : : 3.3 Relative and exact properties : : : : : : : : : : : 3.4 Fiberings : : : : : : : : : : : : : : : : : : : : : : 3.4.1 Applications : : : : : : : : : : : : : : : : 3.5 CW-complexes : : : : : : : : : : : : : : : : : : : 3.5.1 Attaching cells and homotopy properties : 3.5.2 Simplicial complexes : : : : : : : : : : : : 3.5.3 Computing fundamental groups : : : : : : v

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vi 3.6 3.7 3.8 3.9

Simplicial and cellular approximation : : : : : : : : : Weak homotopy equivalence is good enough in CW : Exploiting n-connectedness : : : : : : : : : : : : : : Extracting homotopy groups from known bundles : :

4 Homology and Cohomology Theories

4.1 Introduction : : : : : : : : : : : : : : : : : : : 4.2 Homology and cohomology theories : : : : : : 4.3 Deductions from the axioms : : : : : : : : : : 4.3.1 Reduction and unreduction : : : : : : 4.3.2 Deductions from homology : : : : : : 4.3.3 The Lefschetz theorem : : : : : : : : : 4.4 Homology of chain complexes : : : : : : : : : 4.4.1 Universal coecient theorems : : : : : 4.5 Homology and cohomology of CW-complexes

5 Examples in Homology and Cohomology 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12

Cubical singular homology : : : : : : : : Simplicial singular homology : : : : : : Cup product : : : : : : : : : : : : : : : Geometric simplicial homology : : : : : Computing simplicial homology groups : Relative simplicial homology : : : : : : Geometric simplicial singular homology Bordism homology : : : : : : : : : : : : de Rham cohomology : : : : : : : : : : Geometric simplicial cohomology : : : : More on products : : : : : : : : : : : : : C ech cohomology theories : : : : : : : :

6 Sheaf and Spectral Theories

6.1 Some sheaf theory : : : : : : : : : 6.2 Generalization to spectral theories 6.3 Spectral sequences : : : : : : : : : 6.3.1 Review and moral tale : : :

7 Bundle Theory

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7.2 Stabilization : : : : : : : : : : : : : : : : : : : 7.2.1 Linear stabilization : : : : : : : : : : : 7.2.2 Nonlinear stabilization : : : : : : : : : 7.2.3 Linear K-theory : : : : : : : : : : : : 7.3 Homology and cohomology : : : : : : : : : : 7.3.1 The Gysin sequence : : : : : : : : : : 7.3.2 The Wang sequence : : : : : : : : : : 7.3.3 Transgression and the Serre sequence : 7.3.4 The Leray-Hirsch theorem : : : : : : : 7.3.5 Thom isomorphism theorem : : : : : : 7.3.6 Zeeman comparison theorem : : : : : 7.4 Characteristic classes : : : : : : : : : : : : : : 7.5 Nonabelian cohomology : : : : : : : : : : : :

8 Obstruction Theory 8.1 8.2 8.3 8.4 8.5

Preliminary ideas : : : : : : : : : : : : : : : : Eilenberg-Maclane spaces K (; n) : : : : : : : Moore-Postnikov decomposition of a bration Homotopy cofunctors : : : : : : : : : : : : : : Postnikov invariants : : : : : : : : : : : : : :

9 Applications

Those already done : : : : : : : : : Two classical results : : : : : : : : Theorems of Geroch and Stiefel : : The power : : : : : : : : : : : : : : 9.4.1 Piecewise linear structures : 9.4.2 Smoothing PL structures : 9.4.3 Almost-complex structures 9.5 Marcus's theorem : : : : : : : : : : 9.6 Meta structures : : : : : : : : : : : 9.7 Other signatures : : : : : : : : : : 9.1 9.2 9.3 9.4

A Algebra

A.1 Sets and maps : : : : : : A.2 Categories and functors : A.2.1 A triangular view : A.2.2 Limits of diagrams A.3 Groups and actions : : : : A.3.1 Groups : : : : : : A.3.2 Group actions : : :

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CONTENTS

viii

B Topology B.1 B.2 B.3 B.4 B.5 B.6 B.7

Topological spaces : : : : : : : : : : : : Separation properties : : : : : : : : : : : Compactness : : : : : : : : : : : : : : : Paracompactness : : : : : : : : : : : : : Connectedness : : : : : : : : : : : : : : Peano's space- lling curve : : : : : : : : Collected examples on general topology

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C Manifolds and Bundles

C.1 Manifolds : : : : : : : : : : : : : : : : : : : : : : C.2 Tangent spaces : : : : : : : : : : : : : : : : : : : C.3 Calculus on manifolds : : : : : : : : : : : : : : : C.3.1 Summary of formul : : : : : : : : : : : : C.4 Bundles : : : : : : : : : : : : : : : : : : : : : : : C.5 Metrics and connections : : : : : : : : : : : : : : C.5.1 Principal bundles : : : : : : : : : : : : : : C.5.2 Linear connections : : : : : : : : : : : : : C.5.3 Levi-Civita connection : : : : : : : : : : : C.6 Fibered manifolds : : : : : : : : : : : : : : : : : C.7 Systems of connections and universal connections

D Tables of Homotopy Groups D.1 D.2 D.3 D.4 D.5

Spheres : : : : : : : : : : : : : : : : : : : : : : Three special unitary and symplectic groups : : Symplectic groups : : : : : : : : : : : : : : : : Two spin and two exceptional groups, and C P 2 Real Stiefel manifolds : : : : : : : : : : : : : :

E Computational Algebraic Topology Bibliography Index

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List of Figures 0.1 Spaces of the same homotopy type : : : : : : : : : : : : : : : : : : : 0.2 Torus with the graph of a map S1 ! S1 : : : : : : : : : : : : : : : :

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1.1 Mapping cylinder : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

Deforming graphs of maps relative to a subset A : : : The standard homeomorphism [a; b]  = [0; 1] : : : : : : The torus S1  S1 as an identi cation space : : : : : : Homotopy equivalent but not homeomorphic : : : : : Mapping cylinder Mf and mapping cone Cf of f : : : Deforming circle maps f : S1 ! X : : : : : : : : : : : Join of X to a point  : : : : : : : : : : : : : : : : : : Mapping cylinder bration over its cone : : : : : : : : Cylinder and Mobius line bundles over S1 : : : : : : : Mapping cone Cf for f : X ! B in Top : : : : : : : : Mapping cylinder of  : U ! V : : : : : : : : : : : : : Co ber of co bration q : A ! X : : : : : : : : : : : : Graph of a loop  in loop space Y = Y S1 : : : : : : Suspension of X : SX = S1 ^ X = (S1  X )=(S1 _ X ) :

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Mapping cylinder Mf and mapping cone Cf of f : : : : Schematic ber bundle : : : : : : : : : : : : : : : : : : : Rolling up the real line into a circle : : : : : : : : : : : : Sphere S2 as a CW-complex and as a simplicial complex In nite bouquet of circles embedded in the plane : : : : The torus S1  S1 as a CW-complex : : : : : : : : : : : Deformation retract of attached cell : : : : : : : : : : : Convex coordinates: (; (1 )) : : : : : : : : : : : : : Standard 2-simplex and homeomorphism onto B 2 : : : : Tetrahedron simplicial complex : : : : : : : : : : : : : :

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ix

LIST OF FIGURES

x 3.11 3.12 3.13 3.14

Barycentric subdivision and boundary operator : : Some minimal 2-dimensional simplicial complexes : Non-triangulation of the cylinder : : : : : : : : : : Pseudotriangulation of S1 : : : : : : : : : : : : : :

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i 4.1 Mapping cone of inclusion A ,! X : : : : : : : : : : : : : : : : : : : 112 4.2 S1  S1 as a CW-complex with 1-skeleton S1 _ S1 : : : : : : : : : : : 138 4.3 Projective plane as a CW-complex with 1-skeleton S1 : : : : : : : : : 140

6.1 6.2 6.3 6.4

Picturing spectral sequences : : : : : : : : : : : : : : : : : : : Example of a sparse Ep2 q diagram : : : : : : : : : : : : : : : : Part of the E2 -plane; groups (top) and generators (bottom) Part of the E4 -plane : : : : : : : : : : : : : : : : : : : : : :

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Principal Z2-bundles over S1 : : : : : : : : : : : : : : : Total spaces of associated bundles over S1 with ber B 1 Diagram in the proof of Lemma 7.3.8. : : : : : : : : : : Diagram in the proof of Leray-Serre. : : : : : : : : : : :

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8.1 Postnikov tower of a space : : : : : : : : : : : : : : : : : : : : : : : : 263 8.2 Moore-Postnikov decomposition of a bration : : : : : : : : : : : : : 269 8.3 Fibration used in de ning the nth obstruction set of f : : : : : : : : 276

Preface We have tried to design this book for both instructional and reference use, during and after a rst course in algebraic topology aimed at users rather than developers; indeed, the book arose from such courses taught by the authors. We start gently, with numerous pictures to illustrate the fundamental ideas and constructions in homotopy theory that are needed in later chapters. A certain amount of redundancy is built in for the reader's convenience: we hope to minimize ipping back and forth, and we have provided some appendices for reference. The rst three are concerned with background material in algebra, general topology, manifolds, geometry and bundles. Another gives tables of homotopy groups that should prove useful in computations, and the last outlines the use of a computer algebra package for exterior calculus. Our approach has been that whenever a construction from a proof is needed, we have explicitly noted and referenced this. In general, we have not given a proof unless it yields something useful for computations. As always, the only way to understand mathematics is to do it and use it. To encourage this, Ex denotes either an example or an exercise. The choice is usually up to you the reader, depending on the amount of work you wish to do; however, some are explicitly stated as (unanswered) questions. In such cases, our implicit claim is that you will greatly bene t from at least thinking about how to answer them. Others are explicitly stated directions to write something out: here we are claiming that doing so is essentially necessary for a reasonable understanding of the computational machinery. Those appearing as bald statements of fact may be safely taken as such, but we urge you to play with at least a few of them. We are indebted to all of the books that we cite, and to the many authors and lecturers who have stimulated our enthusiasm for this beautiful and powerful branch of mathematics. It is currently at the forefront of exciting new developments in geometry and physics, perhaps oering the best means yet of describing the fundamental models of physics through topological quantum eld theories. In preparing the manuscript we have used LaTEX and Mathematica tm for generation of PostScript tm graphics, on NeXT tm and OS/2 tm workstations. The title picture is a Mathematica representation of an embedding of a Klein bottle, with self-intersection a gure of eight. For assistance with typing, we are grateful to Kelly Chan, Frances and Chris Dodson, and Karin Smith. Several generations of students at Lancaster University and Wichita State University provided useful comments on xi

xii earlier drafts; in particular we wish to thank B. Bolton, G. J. Fox, M. R. Hanson, H. Oloomi, M. S. Patel, P. D. Sinclair, and C. Snyder for detailed commentary and proofreading. Finally, we wish to thank our good friend David Larner of Kluwer for his encouragement and patience during our preparation of this book, which he accepted in draft form ten years ago. Kit Dodson, Manchester Phil Parker, Wichita October 1, 2000

Introduction and Overview Holeistic mathematics

Topology provides a formal language for qualitative mathematics whereas geometry is mainly quantitative. Thus, in topology we study relationships of proximity or nearness, without using distances. A map between topological spaces is called continuous if it preserves the nearness structures. Now, in algebra we study maps that preserve product structures, for example group homomorphisms between groups, and one of the largest areas of growth in pure mathematics this century has been the solution of topological problems by casting them into simpler form by means of groups. The theory is called algebraic topology and, like analytical geometry and dierential geometry before it, there is considerable interplay with some of the most fundamental theories in physics. In this book we shall develop the essential mathematics and see how it is used to solve problems in geometry and theoretical physics. The fundamental concept is that of homotopy, and arguments based on it have led to some of the deepest theorems in all mathematics, particularly in the algebraic classi cation of topological spaces and in the solution of extension and lifting problems. We shall meet the formal de nitions later but the intuitive idea is very simple:

 Two spaces are of the same homotopy type if one can be continuously deformed

into the other; that is, without losing any holes or introducing any cuts. For example, a circle, a cylinder and a Mobius strip have this property (cf. Figure 0.1), as do a disk and a point. So, coming from geometry, general topology or analysis, we notice immediately that the homotopy relationship transcends dimension, compactness and cardinality for spaces.

 Two maps are homotopic if the graph of one can be continuously deformed

into that of the other. For example, the graphs of maps from a circle to itself lie on the surface of a torus and circuit once the horizontal copy of S1; as indicated in Figure 0.2. Two such maps will be homotopic if they circuit the vertical copy of S1 the same number of times|then they have also the same degree, of course. 1

2

INTRODUCTION

Figure 0.1: Spaces of the same homotopy type

Figure 0.2: Torus with the graph of a map S1 ! S1

AND OVERVIEW

3

Thus, for spaces and maps, the classi cation up to homotopy equivalence precisely captures their qualitative features. Homotopy yields algebraic invariants for a topological space, the homotopy groups, which consist of homotopy classes of maps from spheres to the space. Continuous maps between spaces induce group homomorphisms between their homotopy groups; moreover, homotopic spaces have isomorphic groups and homotopic maps induce the same group homomorphisms. It is possible to obtain information about homotopy groups by means of algebraic theories called homology and cohomology; also, these theories have intuitive appeal and they are computationally simpler. For example, a reasonable space of dimension n can have at most n + 1 (ordinary) homology (and cohomology) groups. Again these groups are homotopy invariants, as are the homomorphisms among them that are induced by continuous maps. Our viewpoint will be that homotopy theory is the basic tool for investigating spaces, while homology and cohomology theories provide useful approximations which are easier to compute and can be tailored to exploit particular features and situations, such as in the study of bundles. One fundamental problem in topology is that of extending a map from a subspace to (be continuous on) the whole space; another is that of lifting a (continuous) map to take values in some overspace, like a covering space or bundle. It turns out to be sucient for many purposes and applications to consider merely homotopy classes of solutions to such problems. This will be our approach and we shall see that a wide range of problems in mathematics and physics can be cast into one of the above two forms. Such problems have led to obstruction theory and its supporting techniques, yielding elegant algebraic objects that encapsulate the reasons for absence of solutions in some cases. We shall use this theory as a higher level tool in the classi cation of spaces relative to their admission of solutions to extension and lifting problems. Our objective is to provide a users' guide to algebraic topology that would allow readers to use the material and results with con dence, in their own applications. Roughly, we suppose in our users a working knowledge of elementary topology and group theory. For your convenience, there is a series of Appendices at the end of the book, covering notation, basic de nitions and results on sets, maps, categories, functors, point set topology, groups, manifolds, bundles, and elds. These sections are intended to help bridge some of the inevitable lapses of memory that so often arise just as a new construction or method seems within our grasp. They also provide a guide to accessible texts on the various topics. The approach is more intuitive than formal and well provided with examples in the places where experience tells us that they are most important. In particular, a good grasp of general topology is dicult to obtain without a wide range of examples and we give special prominence to this. Since any guide needs a fund of data we also append a fairly extensive tabulation of homotopy groups for commonly occurring spaces. The organization of the book is such as to provide motivation for constructions and de nitions and to instill con dence for use of deep theorems by means of worked examples; often we give more prominence to these than to details of proofs. We have tried to avoid repeating standard material in a form that is easily found elsewhere. Since a good way to assess a body of knowledge is to discover what are the questions

4

INTRODUCTION

to which it gives answers, we provide at the end of the next chapter a list of problems from mathematics and physics that are solved by the material contained in the rest of the book. These are all problems of existence or classi cation, but is there any other kind? The next chapter sets the scene by presenting all interesting existence problems for maps fundamentally as extension and lifting problems, up to homotopy equivalence.

Chapter 1

Basics of Extension and Lifting Problems To boldly go where no map has gone before

1.1 Existence problems We begin with some metamathematics. All problems about the existence of maps can be cast into one of the following two forms, which are in a sense mutually dual. The Extension Problem Given an inclusion A ,!i X , and a map A !f Y , does there exist a map f y : X ! Y such that f y agrees with f on A? We shall follow a common practice and indicate the postulation of such a problem by means of the diagram

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Extension Problem

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Here the appropriate source category for maps should be clear from the context and commutativity through a candidate f y is precisely the restriction requirement; that is, f y : f y  i = f y jA = f : If such an f y exists1 , then it is called an extension of f and is said to extend f . In any diagrams, the presence of a dotted arrow or an arrow carrying a ? indicates a pious hope, in no way begging the question of its existence. Note that we shall usually omit  from composite maps. 1

y suggests striving for perfection, crusading

5

CHAPTER 1. BASICS OF EXTENSION AND LIFTING PROBLEMS

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The Lifting Problem Given a pair of maps E !p B and X !f B, does there exist a map f  : X ! E , with pf  = f ? Diagrammatically, the problem is posed thusly: B

p o

O

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f

E

Lifting Problem

? f

X If such an f  exists2 , then it is called a lifting of f and f  is said to lift f . That all existence problems about maps are essentially of one type or the other from these two is seen as follows. Evidently, all existence problems are representable by triangular diagrams and it is easily seen that there are only these six possibilities: ?

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?

6

Of these, 1 and 2 are indistinguishable, as are 3 and 4, but 5 and 6 are reducible by composition to  

and

?



?

5

-

6

Hence 5 has a trivial solution while 6, when well-posed, is simply the existence of a section of a surjection and that is equivalent to the Axiom of Choice. So we are left with just two non-trivial, distinct cases: ?

?

1

and

 6 

?

4

and, plainly, 4 is the lifting problem. We need only show that 1 always converts to an inclusion situation and hence to an extension problem. This is achieved (up to 2

 suggests enlightenment, beati cation

1.1. EXISTENCE PROBLEMS

7

A

I

A A A gA

A A AAU

gA Mg W Figure 1.1: Mapping cylinder homotopy) by the substitution g

A

/

} }} } f }} h } }

Y 

~

W

7 !

i0

A

/

}} } } f }} f y } }

Y 

Mg

~

Here, h = f y jW , and i0 (a) = (a; 0) de nes the natural inclusion of A in Mg which is the mapping cylinder of g (think of A = S1 as in Figure 1.1), Mg = (A  I ) [g W ; where [g means identi cation of (a; 1) with g(a) in the disjoint union of A  I with W. In algebraic situations we are of course solving extension problems when we use a basis or de ne a homomorphism by requiring linearity on generators. The main result from functional analysis is the Hahn-Banach Theorem: Let A be a vector subspace of a Banach space X and let p be a seminorm on X . Then any linear functional

CHAPTER 1. BASICS OF EXTENSION AND LIFTING PROBLEMS

8

f de ned on A with jf (x)j  p(x) can be extended to a bounded linear functional f on X with jf(x)j  p(x). 

From this result it follows that a bounded linear functional on a closed vector subspace A of X can be extended to a bounded linear functional on X , with the same norm.

1.2 Retractions We begin with the simplest case of the extension problem. The Retraction Problem Given an inclusion i : A ,! X and the identity map 1A : A ! A, does there exist a map 1yA : X ! A such that 1yA (called a retraction of X onto A) agrees with 1A on A?

A

o



1A

A 

i /

1yA

X

Retraction Problem



In consequence of the properties of continuous maps we have the following easy results in topological categories:

 There is no retraction of a connected space X onto a disconnected subspace A.

 There is no retraction of a compact space X onto a noncompact subspace A.  There is no retraction of a Hausdor space X onto a non-closed subspace A.  There is a retraction of a space X onto a subspace A, if and only if for all spaces Y every map f : A ! Y admits an extension to X .  Retractions are preserved in products.

1.3 Separation We observe next that the separation properties of the spaces involved will in uence existence problems in topological categories:

 If Y is a T2 space then (and only then) limits are unique, so if A is dense in X it follows that f : A ! Y has at most one continuous extension f y : X ! Y .  Even if A is dense in X , Hausdorness is not sucient to guarantee the existence of a continuous extension. For example, consider

f : R n f0g ! R : x 7 ! 1=x :

1.4. TRANSCRIPTION OF PROBLEMS BY FUNCTORS

9

 We can improve the separation properties in X to be T3, but still we cannot be sure that there are any non-constant continuous real functions at all.

 If Y is a regular space and A is dense in X , then a good enough continuous f : A ! Y has a continuous extension f y to X ; and in this case, since regular implies T2 , the extension is unique. Here, f is good enough if and only if for all x in X and for all neighborhoods Nf (x) of f (x), there exists a neighborhood Nx of x such that

f (Nx \ A)  Nf (x) ; and then f y is the ( lterbase) limit of f . For more details of this aspect of (generalized) convergence see Dugundji [34].

 If X is a T4 space, then we know that there exist non-constant continuous real functions on it. However, subspaces and products of T4 spaces need not be T4.

 If and only if X is a normal space, then every continuous real function on a closed A  X admits a continuous extension to X . This is the Tietze Extension Theorem.

A

o



/

uu uu u uu 9 f y uu u u

f

R

closed



normal space

Tietze Extension Theorem

z

 A Hausdor space X is T4 if and only if given two disjoint nonempty closed subsets A0 and A1 , there always exists a continuous extension to X of

x 7 ! 0 x 2 A f : A0 [ A1 ! [0; 1] : x 7 ! 1 x 2 A0 1

This is the Urysohn Lemma, best remembered as an extension result.

1.4 Transcription of problems by functors The principal role of algebraic topology is the transcription of topological problems concerning existence and classi cation into more tractable algebraic contexts. Transcriptions are eected functorially between a topological category of interest and an appropriate algebraic category. It follows from the way in which diagrams must be preserved by functors that there is a simple necessary condition for existence problems.

10

CHAPTER 1. BASICS OF EXTENSION AND LIFTING PROBLEMS

Let T be a topological category (typically Top ) and let F be a functor from T to a category G . Then, in order to solve in T the problem i

A f

/

X

?

Y 

>

?

or

X

~

E p

B 

f /

it is necessary to be able to solve in G , respectively, the problem F(A) F(i) F(X )

F(E )

/

F(f )

?

F(Y ) 

|

0 for i = 1; 2, whence they both lie in A3 because we have a cover.

21. 22. 23. 24. 25.

26.

27.

28.

29.

So, wrapping a soccer ball with three pieces of paper will force one of the pieces to cover a pair of antipodal points. (See page 122.) Do you want to know if a map has xed points? The Lefschetz number f of a continuous self-map f of a space with nite homology is the obstruction to f being xed-point free. (See page 123.) How many groups act freely on even-dimensional spheres? Only two, because maps on these spheres homotopic to the identity are not xed-point-free. (See page 125.) Which closed surfaces admit a xed-point-free map homotopic to the identity? Only the Klein bottle and torus, because only these have Euler characteristic zero. (See page 125.) One might regard homology as an approximation to homotopy. If so, then how good is it? Well, it's excellent|as far as it goes! But that's just so far, and no farther. (See page 141.) One general principle of algebraic topology is that the more algebraic structure a map must preserve, the easier it is to prove that certain maps do not exist. In this sense, cohomology is better than homology because it has a natural ring structure rather than merely a group structure. The spaces S4  S2 and C P 3 have the same homology and cohomology groups, but dierent cohomology rings. (See pages 150 et seq.) The Euler-Poincare, Hopf Trace, and Lefschetz Fixed Point Theorems: the beauty of these results lies in the fact that elementary linear algebra on chain complexes yields homotopy invariants without any need to pass to homology. (See pages 162 and 163.) The natural process of local integration of forms on chains in Rn ts together among charts on smooth manifolds and is a homotopy type invariant, giving the de Rham isomorphism between cohomologies of dierential forms and chain complexes on compact manifolds. This manifests itself in perhaps the most widely applied result in integral calculus, Stokes's theorem. (See page 168.) On paracompact Hausdor spaces, such as smooth manifolds, there is a natural way to use locally nite covers to generate an abstract simplicial complex, and by the re nement ordering of covers we can take a limit in the induced cohomology. This turns out to give a theory with (at least) dual personalities, C ech cohomology, in which cocycles consist of transition functions between charts. (See Sections 5.12 and 7.1.) Some of our problems require more-or-less ordinary cohomology theories, but with coecients that may vary from point to point (see page 278). Sheaves provide a natural means of allowing this, being algebraical vertically and topological horizontally. (See Section 6.1.)

14

CHAPTER 1. BASICS OF EXTENSION AND LIFTING PROBLEMS

30. All ordinary (co)homology theories agree on CW-complexes. This is indubitably one of the most beautiful applications of spectral sequences, amply rewarding the eort required to cope successfully with them. (See page 196.) 31. Three identical particles moving in a manifold M have a con guration space given by the product M  M  M , less all its diagonals, quotiented by the symmetric group S3 . Selig used spectral sequences to nd the cohomolgy of this space for various M . (See pages 200 et seq.) 32. The infamous complexity of spectral sequences can be somewhat amusingly illustrated by comparison with bureaucracies. It is not at all clear which comes o better! (See pages 204 et seq.) 33. Fiber bundles are extremely important, fundamental objects in both topology and geometry: they are the most useful interpretation of the concept of a parametrized set of spaces. In gauge formulations of quantum eld theories, they are the quantum elds on spacetimes. We nd them conveniently seen as elements of a cohomology theory, especially for computations. (See page 213.) 34. Reducing the structure group from GL(n) to O(1), using the determinant map, yields a reduction to a Z2-bundle. This gives a neat way to describe orientability of vector bundles. (See page 214.) 35. Do all manifolds admit Riemannian metrics? How about Lorentzian metrics, or even general pseudoriemannian metrics of any signature? It took quite a while to gure out that this was a genuine problem, and the complete solution is not yet in sight. We know what we should do, but the actual doing of it seems to be beyond our computational abilities now. (See pages 220, 291, and 292.) 36. Fibrations are surjections that have the homotopy lifting property; so they usually have homotopy-equivalent bers. Fiber bundles are locally products, so they usually have homeomorphic bers. What's the dierence? Fibrations are ber bundles up to homotopy. (See page 236.) 37. The Leray-Serre Theorem: bundles and brations are generalized product spaces, and (co)homology theories can measure the extent of their deviation from a product. Naturally, this involves a spectral sequence. (See page 239.) 38. A CW-complex with just one nontrivial homotopy group  in dimension n is called a K (; n). So S1 is a K (Z; 1), but it is C P 1 , not S2, that is a K (Z; 2). Also, RP 1 is a K (Z2; 1). This highlights some subtleties behind the view that homology groups might give approximations to homotopy groups! (See pages 261 et seq.) 39. When is a manifold orientable? Someone probably asked this before Riemann had nished trying to de ne manifolds. Answering it in the 1930s was one of the original motivations for the entire theory of characteristic classes. (See page 285.)

1.5. THE SHAPE OF THINGS TO COME

15

40. When can a manifold have a spin structure? This is of great interest to physicists, of course; any attempt to introduce quantum theory requires the ability to discuss spin. So their version might read: where can we do physics if we wish to be able to talk about spin? (See page 286.) 41. Which 3- and 4-manifolds are parallelizable? The basic question is classical, but gained currency in modern general relativity when it was asked about globally hyperbolic spacetimes. Many relativists still regard them as the fundamental objects of study. (See page 287.) 42. Which manifolds can be piecewise linear? These are somewhat better than merely topological manifolds, but not as nice as dierentiable manifolds. (See page 288.) 43. Which manifolds are smooth? This has turned into one of the most intriguing questions in dimension 4; in all other dimensions it has been essentially answered. So 4-dimensional topology is now the \hottest brand going." (See page 288.) 44. Which manifolds are complex? The rst part of the answer comes from obstruction theory, but the last part is analytical. For example, all orientable 4-manifolds admit an almost-complex structure, but not all these can be integrated to complex structures. (See page 289.) 45. When does a manifold have any nonvanishing vector elds? Hopf originally answered this apparently simple question; many mathematicians have contributed to solving the multitude of problems it spawned. A compact manifold has a nonvanishing vector eld if and only if its Euler characteristic vanishes; noncompact manifolds always have nonvanishing vector elds, because we can push any problems (zeros) o to in nity. (See page 290.) 46. Which manifolds admit a metalinear structure? How about metaplectic? These questions were of little mathematical interest until the ideas around geometric quantization became important, largely through contact with physics. (See page 291.)

Chapter 2

Up to Homotopy is Good Enough A log with nine holes | old Turkish riddle for a man

2.1 Introducing homotopy In a topological category, a pair of maps f; g : X ! Y which agree on A  X is said to admit a homotopy H from f to g relative to A if there is a map X  I H! Y : (x; t) 7 ! Ht (x) with Ht (a) = H (a; t) = f (a) = g(a) for all a 2 A, H0 = H ( ; 0) = f , and H g (rel A). If A = ? or A is clear from the H1 = H ( ; 1) = g. Then we write f  H g , or context (such as A =  for pointed spaces, cf. below), then we write f  sometimes just f  g and say that f and g are homotopic. We can also think of H as either of:  a 1-parameter family of maps fHt : X ! Y j t 2 [0; 1] g with H0 = f and H1 = g ;  a curve cH from f to g in the function space Y X of maps from X to Y cH : [0; 1] ! Y X : t 7 ! Ht : We call f nullhomotopic or inessential if it is homotopic to a constant map. Intuitively, we picture H as a continuous deformation of the graph of f into that of g; as suggested by Figure 2.1. The following is an easy exercise. Proposition 2.1.1 For all A  X ,  (rel A) is an equivalence relation on the set of maps from X to Y which agree on A.  Maps in the same equivalence class of  (rel A) are said to be homotopic (rel A). 17

CHAPTER 2. UP TO HOMOTOPY IS GOOD ENOUGH

18

X Y

Y

   f 

  P  PP  PP H H PP   PP

g

    

PP PP P

PP PP PP

A

X

Figure 2.1: Deforming graphs of maps relative to a subset A

Ex 1. Supply the proof that  determines an equivalence relation. 2. Use the standard homeomorphism depicted in Figure 2.2

h : [a; b] ! [0; 1] : s 7 ! sb aa to show that

f : [0; 1]

8 2s > < 1 s 2 [01; 141] s2[ ; ] ! [0; 1] : s 7 ! > s + 4 : (s + 1)=2 s 2 [ 41 ; 1]2 2

is homotopic to the identity on [0; 1]. Deduce that being homotopic is a transitive relation on paths and on loops in any space. Observe that a loop in X is a path c : [0; 1] ! X with c(0) = c(1) ; so for loops we are interested in homotopy relf0; 1g. Two topological spaces X , Y are said to be of the same homotopy type or homotopy equivalent if there exist (continuous) maps f : X ! Y; g : Y ! X with gf  1X and fg  1Y :

2.1. INTRODUCING HOMOTOPY

19

1



0  a

  

    



   

  

h : s 7! sb aa

b

Figure 2.2: The standard homeomorphism [a; b]  = [0; 1] Then we write X ' Y and say that f and g are mutual homotopy inverses or inverse up to homotopy. Similarly to the case for maps, ' is an equivalence relation on any collection of topological spaces and one sometimes speaks (loosely) of spaces in the same class as being homotopic.

The spaces in the homotopy equivalence class determined by a singleton space are called contractible; we often use  to denote a singleton space.

Ex

1. Consider the identity map, 1S1 : S1 ! S1, as a closed curve on the torus S1  S1 in Figure 2.3 and nd explicitly two other closed curves on the torus such that all three belong to dierent homotopy classes. 2. If two continuous maps f; g : X ! Sn have f (x) 6= g(x) for all x 2 X then f and g are homotopic. For, otherwise consider tf + (1 t)g : ktf + (1 t)gk 3. Any two continuous maps into a contractible space are homotopic. 4. Show that the following X; Y are homotopically equivalent spaces which are not homeomorphic in the usual topologies (cf. Figure 2.4). Here, S1 _ S1 is the quotient of the disjoint union of two circles, obtained by identifying one point in each circle to each other.

CHAPTER 2. UP TO HOMOTOPY IS GOOD ENOUGH

20

--

S1 6



  

  

   6

1S1

--

S1

Figure 2.3: The torus S1  S1 as an identi cation space

 S1 _ S1 --

6



--

6

--

Punctured torus

6



6



Punctured Klein bottle

Figure 2.4: Homotopy equivalent but not homeomorphic

2.1. INTRODUCING HOMOTOPY (a) (b) (c) (d) (e) (f)

21

X = Sn, Y = Sn  Rm ; X = Rn , Y = f0g; X = Sn 1, Y = Rn n 0; X = S1 _ S1, Y = punctured Klein bottle; X = S1, Y = punctured RP 2 ; X = S1 _ S1, Y = punctured torus.

The objective in this chapter is to assemble a body of theory that exploits our view of homotopy as a fundamental property and brations and co brations as fundamental structures. We can achieve more elegance and power in the results by limiting our attention to Top = ktop, the category of compactly generated Hausdor spaces (also called k-spaces) and continuous maps. This is not too serious a restriction as far as applications are concerned. Certainly, Hausdorness (uniqueness of limits) is normally required in physical spaces and there is a nice functor k (for Kelley), which is a retraction of the category top of Hausdor spaces onto ktop. Thus, any Hausdor X has a compactly generated correspondent k(X ); moreover, this correspondence preserves products and exponentiations (cf. Gray [38], chapter 8). A Hausdor space is called compactly generated if every subset which intersects every compact set in a closed set is itself closed; this property is implied by local compactness or by rst countability (suciency of sequences), for example. Additionally, with a view towards typical applications, we may as well assume that all of our spaces are paracompact (see Appendix B.4). Frequently, we work in Top , the category of pointed spaces and pointed (that means base-point preserving) maps, where homotopies are always understood to be relative to the set consisting of the base point . Then [(X; ); (Y; )], or brie y [X; Y ], is used to denote the set of homotopy equivalence classes [f ] of pointed maps f from (X; ) to (Y; ). Two pointed spaces (X; ) and (Y; ) are called of the same homotopy type or homotopy equivalent if there are pointed maps p and q with [p] 2 [X; Y ] and [q] 2 [Y; X ] which are mutual homotopy inverses, namely such that [qp] = [1X ] and [pq] = [1Y ] : Then we write X ' Y , and it is easy to show that ' is an equivalence relation on any collection of pointed spaces. The singleton pointed space (; ), often denoted simply by , de nes the class of contractible pointed spaces. In Top there is only one constant map from (X; ) to (Y; ), that which sends all of X to  2 Y . We shall use the notation f   to mean that f is homotopic to the constant map. It follows that (X; ) is contractible precisely when 1X  .

Ex on homotopy (Work in Top.) 1. A circle S1, a cylinder S1 I, and a solid torus S1 B 2 are mutually homotopic.

CHAPTER 2. UP TO HOMOTOPY IS GOOD ENOUGH

22

f :X !Y

X

Mf

Y

Cf

Y Figure 2.5: Mapping cylinder Mf and mapping cone Cf of f

2. Euclidean spaces Rn , unit balls B n , and unit cubes In are contractible. 3. Is contractibility preserved in products, quotients, restrictions and for retractions? 4. Any two maps into a contractible space are homotopic and hence all are homotopic to the constant map. What about maps from contractible spaces? 5. If any f : X ! Sm is not surjective, then it is nullhomotopic. 6. A map f : X ! Y is nullhomotopic if and only if it extends to a map from the cone over X : CX = (X  I)=(X  f1g) : Note that C Sn 1  = B n. 7. Given a map f : X ! Y , then Y is homotopy equivalent to the mapping cylinder of f , Figure 2.5,

Mf = (X  I) [f Y = (X  I t Y )= where for all x in X ,  identi es (x; 1) with f (x). Details of a suitable homotopy can be found in Hocking and Young [46], p. 157. 8. The mapping cone of f : X ! Y is Cf = Mf =X , Figure 2.5.

2.1. INTRODUCING HOMOTOPY

23

9. A subset A ,! X is called a deformation retract of X if there is a retraction r : X ! A, so rjA = 1A, with r  1X ; then A ' X . In this case every h : A ! Y extends to X . 10. For all f : X ! Y , the space Y is a deformation retract of the mapping cylinder Mf , shown on the left in Figure 2.5. (Cf. Ex 7 above.) 11. A space X is contractible if and only if it is a retract of the mapping cylinder M of any constant map X ! fg. Observe that M is a homeomorph of the cone CX . 12. One may supply appropriate base points and repeat these Ex in Top . Observe that reduced versions of the cone and cylinder are required: these are obtained by identifying all those points fg  I in X  I. Draw some of these schematically. i 13. Given an inclusion A ,! X , then X=A is homeomorphic to (X [i CA)=CA. See Switzer [106, p. 26] for details and for further homotopy results concerning cones.

It is clear that homeomorphic spaces are homotopy equivalent, but not conversely. Hence, by working with homotopy equivalence instead of topological equivalence we lose some detail. This loss is of course irrelevant in the solution of pure existence problems; but, surprisingly, we shall see also classi cations of extra structures on spaces usefully based on homotopy classes. A recurring theme will be to exploit homotopy-type invariants by using the fact that if they dier for two spaces then the spaces cannot be homotopy equivalent and so must dier topologically. Borrowing terminology in anticipation of later structures, we might think of homotopy as yielding a kind of bering of topology theory, with some algebraic structure arising on bers from homotopy-type invariants. A very nice illustration of the role of homotopy in an extension problem is the following, where we consider extending a continuous map from the unit circle S1 to the disk or 2-ball B 2 .

Proposition 2.1.2 (Circle to disk extension) A continuous map from a circle extends to the disk if and only if it is inessential.

S1

o

f



inclusion

fy

X

/

B2 Extension

() f  



z

Motivation: Of course, if X is contractible then we always have f  . So for a mental picture think of X as S1, whence the graph of f lies on

the surface of a torus and f y would have a graph inside the solid torus,

CHAPTER 2. UP TO HOMOTOPY IS GOOD ENOUGH

24 

X f









AK A fy A A

S1



r

c

,! 

B2

Figure 2.6: Deforming circle maps f : S1 ! X agreeing with f on its surface. Then f   if and only if the graph of f can be continuously deformed into the vertical circle c in Figure 2.6,

regarded as the graph of a constant map. Such a deformation is not possible if f gets right around X , as indicated by the horizontal circle r in Figure 2.6. Now, if f   we can obtain f y by taking the constant slice B 2  fg and then reversing its deformation to bring its perimeter back into agreement with f . Conversely, if we are given f y then the graph of f y cannot be like r in the gure, because such a graph could not arise from the continuous image of B 2 . Pictorially, any slice of the torus through a curve like r will contain a hole, since r is not deformable to c, but f y (B 2 ) cannot have such a hole if f y is to be continuous.

Proof: For convenience we consider S1 as the boundary of the unit disk B 2 = fz 2 C j 0  jz j  1g with base point  = 1.

(i) Suppose that we are given f y , continuously extending f , with f (1) = f y (1) =  2 X . Then we obtain a homotopy f   by S 1  I ! X : (ei ; t) 7 ! f y (ei(1 t) ) : (ii) Conversely, given a homotopy H from f to  we construct f y : B 2 ! X : rei 7 ! H (ei ; 1 r) : Now, this has the required restriction because r = 1 on S 1 and f y jS1 = H ( ; 0) = f which completes our proof. 

2.1. INTRODUCING HOMOTOPY

25

In fact, this result extends to spheres of all dimensions:

Theorem 2.1.3 (Sphere to ball extension) Any continuous map from a sphere to the ball it bounds extends if and only if it is inessential. Sn

inclusion

o



f

/

B n+1 Extension

fy

X

() f  



y

Proof: Ex



For the particular case f = 1Sn we have the retraction problem. However, for n  0 it is a fact that S n is not contractible, as we shall see later, though B n is

contractible.

Corollary 2.1.4 There is no retraction of a ball onto its boundary. Proof: Since Sn 6' , then 1Sn 6  and so 1Sn has no extension to Bn+1 .



The next result also concerns S n and is attributed to Borsuk.

Theorem 2.1.5 (Extending maps from closed sets in Rm to spheres) Any continuous map into a sphere from a closed subset of Rm extends to the whole of Rm if and only if it is inessential. A

o



f

Sn

closed

fy

/

Rm Extension

() f  



y

Proof: Hocking and Young [46, p. 53] actually show that if f  g and f y exists, then also gy exists with f y  gy . The result follows because constant maps always extend. Moreover, A can be a closed subset of any separable metric space.  We remark that in order to show f  , we need nd only one homotopy; however, to show f 6  we need to establish that no such homotopy exists, clearly a harder task. We can obtain a useful characterization of inessential maps by the following construction of the join of a space X and a singleton  as illustrated in Figure 2.7.

X = (fg  X  I)=(fg X  f1g) : There is a natural injection: i0 : X ! X : x 7! (x; 0).

CHAPTER 2. UP TO HOMOTOPY IS GOOD ENOUGH

26



X



X

X I

Figure 2.7: Join of X to a point 

Proposition 2.1.6 (Extension to join with singleton) A map extends to the join of its domain with a singleton if and only if the map is inessential. X

o



f

i0 /

X Extension

fy

Y 

() f  

~

Proof: (i) Given f y we nd a homotopy (Check the continuity!)  (x; t) 7 ! f y(x; t) for t 2 [0; 1) X  I ! Y : (x; 1) 7 ! f y () from f to a constant map. (ii) If we have a homotopy H : X  I ! Y from f to a constant map

c : X ! Y : x 7 ! y0 ; then put

f y : X

 (x; t) 7 ! H (x; t) !Y :  7 ! y0

This map has the desired properties.

for t 2 [0; 1)



Observe that implicitly we worked here in Top, rather than in Top which is more dicult by being constrained to the one constant base-point preserving map. Our

2.2. FIBRATIONS AND COFIBRATIONS

27

proof does not carry over directly; for if i0 is a base-point preserving injection, then  is not the base point in X and we get the wrong constant map in (i). However, we can decide precisely when two constant maps are interchangeable, up to homotopy, by the following. Proposition 2.1.7 Two constant maps, ci : X ! Y : x 7 ! yi ; i = 0; 1 ; are homotopic if and only if there is a continuous curve

: [0; 1] ! Y from y0 to y1 :

Proof: (i) Given a homotopy c0 H c1 and any x 2 X we have

x : [0; 1] ! Y : t 7 ! H (x; t) : (ii) Given the curve we obtain a homotopy H : X  I ! Y : (x; t) 7 ! (t) which completes the proof.



2.2 Fibrations and co brations Good Fibrations|B. Boys

We now make a systematic approach to lifting and extending problems via the representation of maps as brations and co brations. These turn out to be very useful concepts and quite widespread in applications because, up to homotopy, every map is equivalently represented as a bration and also as a co bration. We work in Top .

De nition 2.2.1 A (pointed) map p : E ! B is called a bration when it has the homotopy lifting property, namely: every square diagram like that below has a

diagonal H  lifting it.

X j0

g

X I H 

/

E

zz H zzz p zz z z
0 RP n as a CW-complex; similarly C P 1 and H P 1 are CW-complexes.

CHAPTER 3. HOMOTOPY GROUP THEORY

72






[1n=1 f(x; y) j (x






1 2 1 2 n ) + y = n2 g

Product topology

2 2 2 [1 n=1 f(x; y ) j (x n) + y = n g

Weak topology

Figure 3.5: In nite bouquet of circles embedded in the plane 9. (i) Every cell complex contains at least one 0-cell, for if it contains an n-cell then its k-skeleton is not empty for some k < n. X 0 is a discrete space. (ii) A CW-complex is path connected if and only if it is connected. 10. The torus S1S1 is a CW-complex with one 2-cell, two l-cells and a 0-cell, as we can see by displaying the torus as a rectangle in the plane with identi cations as indicated in Figure 3.6. 11. Construct CW-complexes for the cylinder and the Klein bottle. 12. For any n-cell e with characteristic map  in a cell complex X;  (B n ) = e . If a cell complex is nite, then it is compact. If A is a subcomplex of X and e  A, then also e  A. Moreover, A is closed, its induced topology is the weak topology, and X=A is a CW-complex. For each n  0, the n-skeleton X n is a subcomplex of X . 13. For any cell complexes X and Y there is a cell-complex structure for X  Y with product cells and the characteristic maps obtained from the product maps by using homeomorphisms:

B n+m  =Bn Bm: = In  Im  = In+m  14. If X and Y are CW-complexes then X  Y is a CW-complex and X _ Y is a subcomplex. (Cf. Gray [38], p. 117; the compactly generated properties of X and Y are needed here; cf. Maunder [68], p. 282, Switzer [106], p. 72.)

3.5. CW-COMPLEXES

e0

1 e-

e1 ?

e2

e0

73

e0

e2

? e1

e0

--

e1

e1 e1

e0 

Figure 3.6: The torus S1  S1 as a CW-complex 15. Attaching and removing cells If X is a CW-complex with n-skeleton X n , then:

 For any f : Sn ! X n , (X t B n+1 )=fx  f (x) j x 2 @ B n+1 g is a CW-complex denoted X [f en+1 .  For any (n + 1)-cell e , X n+1 n e is a subcomplex and  (X n+1 n e ) [f en+1 X n+1 = for some f : Sn ! X n. 16. Given a CW-complex X with fe ;  j  2 g, then f : X ! Y is continuous if and only if f   is continuous for all  2 .

Theorem 3.5.3 (CW-subcomplex inclusion co bration) The inclusion of a subcomplex A in a CW-complex X is a co bration.

X  f0g h j0

Y uu u u H yuu H u u u uu

X I 

o

/

:

O

CW-subcomplex

co bration

AI

Proof: We construct a sequence of maps for n  0, by induction: Hn : X  f0g [ (X n [ A)  I ! Y

CHAPTER 3. HOMOTOPY GROUP THEORY

74

(cf. Gray [38, p. 118] and, for another approach, Switzer [106], p. 75). The 0-skeleton

X 0 has the discrete topology and so given H and h put:  (e0; t) if t 6= 0 ; H0 : (e0 ; t) 7 ! H h(e0 ; 0) if t = 0 :

Suppose that we have Hn for some n  0; then construct Hn+1 as follows. Since Hn has dealt with all n-cells, consider an (n + 1)-cell e in X n+1. Then X n+1 n e is a subcomplex and for some f : Sn ! X n

X n+1  = (X n+1 n e ) [f en+1

by Ex 14. Also, we can construct the composite

[ B n+1  f0g (!;1) X  f0g [ (X n [ A)  I H!n Y and extend it to y on B n+1  I. Now,  restricts to be a homeomorphism from the interior of B n+1 onto e , for each choice of e , and the boundary of this cell is an image of Sn in X n.  : Sn  I

Thus we de ne the map

Hn+1 : X  f0g [ (X n+1 [ A)  I ! Y

 H (a; t) if (a; t) 2 X  f0g [ (X n [ A)  I ; : (a; t) 7 ! yn(a; t) if (a; t) 2 e  I  = B n+1  I :  

This is continuous, so the induction is complete and our required extension of the homotopy H is

H y : X  I ! Y : (x; t) 7 ! Hn (x; t) if x 2 X n ; which is continuous because it is continuous on every n-skeleton subcomplex of the CW-complex X and X  I is a CW-complex. 

Corollary 3.5.4 Up to homotopy, we can quotient out contractible subspaces: A '  =) (X; A) ' (X=A; fg) : Proof: We show that the projection p : X ! ! X=A is invertible up to homotopy. Given H , we extend it to H y

X

A o

O

O

Hy

X I o

H

AI

3.5. CW-COMPLEXES

75

by the co bration property. Since for all a 2 A

H1 (a) = H1y (a) =  ;

 x if x 62 A

we obtain a map

k : X=A ! X : [x] 7 !  if x 2 A with H1y = kp. Hence kp  1x = H0y . Next, pH y (a; t) =  (8 a 2 A ; t 2 I) ;

so there is a homotopy (cf. Ex 1 below)

G : X=A  I ! X=A with pH y = G(p; 1) : Then

G0 ([x]) = G0 p(x) = G(p(x); 0) = pH y (x; 0) = p(x) :

Hence G0 = 1X=A . Also

G1 ([x]) = G1 p(x) = G(p(x); 1) = pH y (x; 1) = pH1y (x) = pkp(x) : Finally, G1 = pk, since p is surjective, and pk  1X=A .



This means that collapsing contractible subcomplexes does not matter, up to homotopy. Which is how it should be: we only want to synthesize enough matter to show where there are internal holes; there's nothing baroque about CW-complexes. Really of course we only want the holes but they become too much at one with the tao if we don't have their boundaries too. Some practical researchers in another eld demonstrated how to show complex hole patterns in three dimensions. They were studying moles; by ooding a large abandoned molehill with expanding polystyrene they could excavate and reveal the tunnel complex without its bounding earth. Apparently, after making their complex, moles simply patrol their holes and eat the worms that drop in, so perhaps both are really interested in topology|for opposite reasons. As the theory develops you might give thought to whether there are more theorems for moles or for worms. Or wilt thou go ask the mole?|Blake, The Book of Thel

Ex on quotient spaces 1. If A is a closed subspace of X with a homotopy F : X  I ! Y satisfying F (a; t) = F (b; t) for all a; b 2 A; t 2 I, then there is a homotopy G : X=A  I ! Y. 2. The corollary does not require X to be a CW-complex, but A ,! X is required to be a co bration.

CHAPTER 3. HOMOTOPY GROUP THEORY

76

We can now make some reasonable progress in nding homotopy groups for

CW-complexes. Four theorems that we use in the sequel to make very considerable

advances are quite easily understood and can be summarized brie y as follows, for

CW-complexes X; Y :

 (J. H. C. Whitehead) Given f : X ! Y in Top, and any 0-cell e0 in X , then X ' Y if f n (X; e0 ) ! n (Y; f (e0 )) is bijective for all n  0.  (Cellular approximation) Given f : X ! Y in Top, then the problem of extension in CW

K Y 

o



inclusion

/

x xx x  xx y x  xx

X

|

has a solution y  f (rel K ).  (H. Freudenthal) If m (X ) = 0 for m = 0; 1; : : :; n 1, and fg ,! X is a co bration, then the map induced by suspension

S : m (X ) ! m+1 (SX ) is an isomorphism for m  2n 2 and an epimorphism for m = 2n 1.  (Homotopy excision) Given a CW-complex W = X [ Y and m; n;  1 with (X; X \ Y ) n-connected and (Y; X \ Y ) m-connected, then the map induced j by the inclusion (X; X \ Y ) ,! (X [ Y; Y ),

j : r (X; X \ Y ) ! r (X [ Y; Y ) is an isomorphism for 1  r < n + m and an epimorphism for r = n + m. The rest of this chapter is devoted to these results, incidental constructions to their proofs, and applications. We begin with a study of certain relationships between homotopy properties and the attaching of cells to spaces. Then we spend a little time on simplicial complexes and their barycentric subdivisions in order to have at our disposal the useful simplicial approximation theorem.

3.5.1 Attaching cells and homotopy properties

Theorem 3.5.5 (Characterising relatively inessential maps) Given a map of pairs f : (B n ; Sn 1) ! (X; A) then

[f ] = 0 2 n (X; A) () 9 g : B n ! A with f  g(rel Sn 1) :

3.5. CW-COMPLEXES

77

Proof: ()) Given f H  then f G g(rel Sn 1) with G:Bn I

(

for jj x jj2 [0; 1 t=2) ( 1 xt=2 ; t) ! X : (x; t) 7 ! H x H ( jjxjj ; 2 2 jj x jj) for jj x jj2 [1 t=2; 1]

F  with (() Given g, then [g] = [f ]. Hence f 

F : (B n ; Sn 1)  I ! (X; A) : (x; t) 7 ! g((1 t)x + tx0 )

where x0 =  2 B n .



The next result was met previously in x3.1.3; here we prove it.

Theorem 3.5.6 (Only inessential maps extend to interior of cell) For any integer n  1, we have: Sn inclusion vB n+1 o



/

vv vvy v vv g vv

g

X 

Extension gy exists

()

[g] = 0 2  (X ) n

z

Proof: ()) Given gy de ne H : Sn  I ! X : (x; t) 7 ! gy ((1 t)x + tx0 ) where x0 =  2 Sn; then H0 = 1 and H1 = . G  de ne (() Given g   for kxk  1=2 ; gy : B n+1 ! X : x 7 ! xG( x ; 2 2kxk) for kxk > 1=2 : kx k Then, if x 2 Sn we have kxk = 1 so gy jSn(x) = G(x; 0) = G0 (x) = g(x) : Thus we have the extension; it is continuous because it is continuous on the two closed subsets of B n+1 :

fx 2 B n+1 j kxk 2 [0; 1=2] g ; fx 2 B n+1 j kxk 2 [1=2; 1] g and so we may apply the following easy but useful result, Lemma (Gluing Lemma) Let A; B be closed (or open) subsets of A[B. If two continuous maps f : A ! X and g : B ! X agree on A \ B , then their union map f [g :A[B !X is continuous. 

CHAPTER 3. HOMOTOPY GROUP THEORY

78

to nish the proof of the theorem.



Corollary 3.5.7 In the particular case that X ' , we nd that g : Sn ! X extends to B n+1 because every map into a contractible space is nullhomotopic.  Theorem 3.5.8 (Extension to attached n-cell) In the diagram: Sn

1

g /

A

o



f

inclusion /

A [g en

fy

X

Extension exists if



0 (X ) = 0 = n

1

(X )

z

Proof: We know that f extends to A [g C Sn 1 if and only if fg is nullhomotopic. But C Sn 1  = B n , hence A [g en  = A [g C Sn 1 ; so the required f y exists if and only if fg  , which is the case because [fg] 2 [Sn 1; X ] = n 1 (X ) = 0 by hypothesis.



The next result says that attaching n-cells to a CW-complex yields a quotient which is a bouquet of the boundaries of those n-cells.

Theorem 3.5.9 (Attaching cells gives relatively nice bouquet) Let (X; A) be a relative CW-complex. If X is obtained from A by attaching the set of n-cells f ; en j  2  g, then there is a homeomorphism _  : X=A  = Sn and a commutative diagram in Topp :

(Sn; fg)

tt qtttt tt tt

2

i

_ /

 2

9

(B n ; @ B nN)

NNN N NNNNN '

(X; A)

Sn; fg

!

O

=  p /

(X=A; fg)

Here q collapses the boundary of the n-ball to a point, so giving the n-sphere.

3.5. CW-COMPLEXES

79

@B n

@B n



Bn

A [ en



A

:x

Figure 3.7: Deformation retract of attached cell

Proof (outline): Since (X; A) is a relative CW-complex, X=A is a CW complex. This is because if e is an m-cell in A with characteristic map , then B m = e

is also contained in A. Since A contains X n 1 ,  in the diagram is well de ned. Now, by de nition X consists of A with n-cells attached, so when we quotient by A it leaves a bouquet of n-spheres attached to the residue of A, namely . (Recall that B n =@ B n  = Sn.) Thus the upper row in the diagram is rather like a coordinate version of the bottom row.  As usual, we use en to denote the interior of en; see page 311.

Theorem 3.5.10 (Strong deformation retract) If X = A [ en and x is in en, cf. Figure 3.7, then A is a strong deformation retract of X n fxg: ri  1A(rel A) : A

o



/

X nfxg

y yy y yr yy yy

1A

A

i



h

|

Proof: The construction depends on the following strong deformation retract of the punctured n-ball B n n fbg onto @ B n : Hb : B n nfbg  I ! B n nfbg : (b + (1 )z; t) 7 ! (1 t)b + (1 (1 t))z ; where we have used convex coordinates for B n n fbg in terms of the given interior point b 2  B n and boundary points z with  2 [0; 1); cf. Figure 3.8. (These coordinates take their simplest form if b is the origin.) Now, by de nition, there is a characteristic map for en in X ,

 : (B n ; @ B n ) ! (X; A) ;

CHAPTER 3. HOMOTOPY GROUP THEORY

80

x (1 ) 


b 0

z

Bn

@B n

Figure 3.8: Convex coordinates: (; (1 )) and since it is bijective from  B n to en , then b 2  fxg is uniquely determined. Hence we have the required homotopy:

 (a; t) 7 ! a H : X nfxg  I ! X nfxg : (x; t) 7 ! H (y; t) ifif ax 2= A(;y) 62 A : b

 We shall make signi cant use of the simple convex structure of B n to provide a convex linear structure on the interior of an n-cell, via the good properties of its characteristic map. In the notation of the proposition, we use the linear combination

8 x; y 2 en < rx + sy = (r (x) + s (y)) for : (r (x) + s (y)) 2 B n r; s 2 R ;

where it is well de ned, for example when r; s  0 with r + s = 1. Before employing this we need a construction aid to handle simple spaces.

3.5.2 Simplicial complexes De nition 3.5.11 The standard n-simplex is X
n = f(xi ) 2 Rn+1 j xi = 1; xi 2 [0; 1] 8 ig

3.5. CW-COMPLEXES

81

v1 v

2

B

2

2

y x b

v0

Figure 3.9: Standard 2-simplex and homeomorphism onto B 2 and its vertices (cf. Figure 3.9) are the n + 1 points

v0 = (1; 0; : : : ; 0); v1 = (0; 1; 0; : : :; 0); : : : ; vn = (0; : : : ; 0; 1) : Conversely, given n +1 points v0 ; v1 ; : : : ; vn in Rn+1 such that the n vectors fvi v0 j i = 1; : : : ; ng are linearly independent, then they de ne an n-simplex. It is the subset:

(v0 ; v1 ; : : : ; vn ) = fx 2 Rn+1 j x =

X

ti vi ; ti 2 [0; 1];

X

ti = 1g

which we say spans the vertices v0 ; v1 ; : : : ; vn with barycentric coordinates (ti ). Evidently we can equally well consider an n-simplex to be de ned in any Rk with k  n. We often abbreviate (v0 ; v1 ; : : : ; vn ) to  when the vertices are clear from context.

Ex on barycenters 1. There is a homeomorphism
n ! (v0 ; v1 ; : : : ; vn ) : (xj ) 7 !

n X i=0

xi+1 vi :

2. The barycenter (center of gravity) of (v0 ; v1 ; : : : vn ) is the point with the average coordinates, namely b(v0 ;v1 ;:::;vn ) = n +1 1 (v0 + v1 +


+ vn ) : 3. Radial scaling from the barycenter gives a homeomorphism of any n-simplex onto an n-ball and hence onto B n , Figure 3.9.

82

CHAPTER 3. HOMOTOPY GROUP THEORY

Figure 3.10: Tetrahedron simplicial complex 4. Given an n-simplex (v0 ; v1 ; : : : ; vn ), every subset (i0 ; i1 ; : : : ; ik ) of (0; 1; : : : ; n) determines a k-simplex (vi0 ; vi1 ; : : : ; vik ) called a k-face or k-subsimplex of the given n-simplex; ? is a face of every simplex. De nition 3.5.12 A geometric ( nite) simplicial complex K is a ( nite) collection fi 2 Rm j i = 1; 2; : : :; pg of simplices, all in Rm for some nite m, satisfying: (i) i \ j is a face of i and of j (ii) every face of a simplex in K is itself a simplex in K . A simplicial complex K inherits the subspace topology and we denote this topological space by jK j. It is called a realization of K . Then a space X , homeomorphic to jK j is called a polyhedron and we say that K is a triangulation of X . The n-skeleton K n of a simplicial complex K is that subcomplex consisting of m-faces of simplices of K for m  n. By convention, the empty set is the ( 1)-skeleton .

Ex on tetrahedra A tetrahedron is a simplicial complex in R3 with: four 2simplices, six 1-simplices, and four 0-simplices. This is illustrated in Figure 3.10 as a surface in R3 and as the identi cation space in the plane that we rst met in elementary school origami. The opened out display in the plane is a very convenient way to present simplicial complexes that represent surfaces and we shall see more examples below (cf. Figure 3.12). The 1-skeleton of the tetrahedron consists of the six 1-faces and four vertices. The 0-skeleton is simply the vertex set.

For any complex, we always have subcomplex inclusions of its skeletons: ? = K 1 ,! K 0 ,! K 1 ,!


,! K r = K for some r; if K r 1 6= K we say that K is r-dimensional. A map f : jK j ! Rn for a simplicial complex K; is called linear (and this ensures continuity) if for all  = (v0 ; v1 ; : : : ; vn ) 2 K and for all x 2  : X X f (x) = f ( ti vi ) = ti f (vi ) :

3.5. CW-COMPLEXES

83

Figure 3.11: Barycentric subdivision and boundary operator A map between simplicial complexes is called simplicial if it sends vertices to vertices and is linear over linear combinations of vertices. The barycentric subdivision of a simplicial complex K is a simplicial complex K 0 satisfying (cf. Figure 3.11): (i) the vertices b of K 0 are the barycenters of the simplices  of K ; (ii) the simplices of K 0 are

f (b0 ; b1 ; : : : ; bm ) j i  i+1 strictly g :

The boundary @ of a simplex  = (v0 ; v1 ; : : : ; vn ) is the simplicial complex with simplices (cf. Figure 3.11):

f (v1 ; v2 ; : : : ; vn ); (v0 ; v2 ; v3 ; : : : ; vn ); : : : ; (v0 ; v1 ; : : : ; vn 1 ) g : We often denote such sets by means of u^ to indicate the omission of the vertex u:

@ (v0 ; v1 ; : : : ; vn ) = f (v0 ; v1 ; : : : ; v^i ; : : : ; vn ) j i = 0; 1; : : :; n g :

Ex on barycentric subdivision (Cf. Switzer [106] x12 for these and others, for

example.)

1. If K has a barycentric subdivision K 0, then it is indeed unique and a simplicial complex with jK j  = jK 0 j. 2. Every simplicial complex has a barycentric subdivision. 3. A simplex  has a barycentric subdivision if @ has one.

84

CHAPTER 3. HOMOTOPY GROUP THEORY 4. For any n-simplex , indeed @ is a simplicial complex and j@j  = Sn 1. 5. Every simplicial complex is a CW-complex. 6. Given simplicial complexes K and L in Rm , then de ne K \ L to be the set of simplices in both K and L and K [ L to be the set of simplicial complexes in either K or L. Then it follows that K \ L is a subcomplex of K and L but, in general, K [ L is not a simplicial complex. However, K [ L is a simplicial complex if we have

jK \ Lj = jK j \ jLj ; and this latter happens in particular if K and L are subcomplexes of some other simplicial complex. 7. The Euler characteristic (K ) of a nite simplicial complex K ,

(K ) =

dim XK

i=0

( 1)i i

where i is the number of i-simplices in K , is unaltered by barycentric subdivision of K . The historical importance of geometric ( nite) simplices is that they can be used to synthesize homeomorphs of the compact spaces encountered in elementary geometry. This is quite clear intuitively; what is less clear is how the synthesis can be most eciently carried out. However, we shall need to triangulate only simple spaces (particularly spheres and balls) and to construct linear maps on them.

Ex on more origami: triangulations of simple surfaces 1. The following two-dimensional spaces admit the triangulations indicated by identifying the similarly marked edges in the plane diagrams of Figure 3.12. In fact, each of these triangulations is minimal in that there is no alternative with smaller vertex set. For every closed surface S the number, N0 , of vertices in a triangulation satis es   p N0  21 7 + 49 24S where S is the Euler characteristic of the surface. Now, S is a topological invariant and can be calculated from any triangulation as

S = N0 N1 + N2 where N1 is the number of 1-simplices and N2 is the number of 2-simplices.

3.5. CW-COMPLEXES

85

Figure 3.12: Some minimal 2-dimensional simplicial complexes

K

v

u

v

Figure 3.13: Non-triangulation of the cylinder

86

CHAPTER 3. HOMOTOPY GROUP THEORY 2. As examples of non-closed surfaces we have the cylinder and Mobius strip in Figure 3.12. (Observe that we always need at least three 1-simplices to `get round' any copy of S1). Again, these two triangulations are minimal. If we try to eliminate some of the 2-simplices then we nd that there arise two distinct 1-simplices between one pair of vertices, or worse, a 1-simplex is doubled back on itself, as in Figure 3.13. There is evidently no doubt that this latter is a homeomorph of the cylinder. What fails is the simplicial complex property:

;  2 K =)  \  = ? or  \  is a common face. For example, in the collection of simplices joined as indicated in Figure 3.13, consider the intersection of the two 2-simplices ;  ,

 \  = fu; vg : But the only face in  arising from this pair of vertices is

v ! u and the only face arising from it in  is u ! v : Hence, fu; vg is not a common face of  and  , so K is not a simplicial complex;

although it is a homeomorph of the cylinder, it does not yield a triangulation but a pseudotriangulation. In fact, pseudotriangulations yield similar algebraic invariants to triangulations (cf. Hilton and Wylie [44], pp. 50,132{3), but they are less convenient in the combinatorial development of the theory. The essential dierence is that in a pseudotriangulation we may have more than one simplex spanned by a given set of vertices, as both  and  are spanned by fu; vg in the example; this is not allowed in a triangulation. So we might call an entity like K in the example of Figure 3.13 a pseudosimplicial complex. It turns out that the rst barycentric subdivision of a pseudosimplicial complex is actually a simplicial complex. Hence, every pseudotriangulation determines a unique triangulation; but it need not be a minimal triangulation. For example, in Figure 3.14, K is a pseudotriangulation of S1, but its rst barycentric subdivision, the triangulation K 0 in 3.14, has one more vertex than the minimal triangulation|which is shown at the lower right in Figure 3.11.

3.5.3 Computing fundamental groups As intuition suggests for polyhedra, their fundamental groups are computable from a study of the loops which are closed paths along 1-simplices, so-called edge loops. Evidently, homotopy will discard any parts of such paths which are part of the boundary of any 2-simplex. More precisely, an edge loop at the vertex v in a

3.5. CW-COMPLEXES

87

Figure 3.14: Pseudotriangulation of S1 simplicial complex K is a sequence of vertices v; u; : : : ; w; v, and we identify two such sequences

v; u; a; b; c; : : : ; w; v

and

v; u; a; c; : : : ; w; v

if abc span a 2-simplex, a 1-simplex, or a vertex of K . Then we de ne the edge group of K at v, E (K; v), to be the group formed from equivalence classes of edge

loops at v with the binary operation of juxtaposition.

Ex 1. Find E (K; v) when K is a triangulation of a cylinder. 2. Satisfy yourself that E (K; v) is always a well-de ned group and that there is a group isomorphism from it to 1 (jK j; v). Full details can be found in Armstrong [2], pp. 133{134. It is convenient to present the edge group as a set of generators and relations. We can do this as follows; again full details are to be found in Armstrong [2], pp. 134{ 136. Let L be a simply-connected (cf. x2.5) subcomplex of K which contains all of the vertices of K . De ne G(K; L) to be the group generated by the 1-simplices of K , denoted gij for each edge vi ; vj and with relations:

gij = 1 if vi ; vj span a simplex of L gij gjk = gik if vi ; vj ; vk span a simplex of K : To each generator gij we can associate an edge < vi ; vj >. Next, because L is path-connected, we can construct an edge path ek from v to each vk and its inverse ek 1 from vk to v. Hence we obtain a map which turns out to be an isomorphism of groups:

G(K; L)  = E (K; v) :

CHAPTER 3. HOMOTOPY GROUP THEORY

88

Ex 1. Satisfy yourself that the above construction works and nd G(K; L) for the case when jK j is a one-point union of two circles and a disk. 2. In computing 1 (jK j; v) as G(K; L), why is it best to choose L as large as possible? Show that a suitable L always exists, as the maximal tree in K , but that it can often pro tably be enlarged to simplify calculations by choosing a larger subcomplex that is simply connected. 3. Compute the fundamental groups of a torus and a Klein bottle. The next theorem allows us to compute fundamental groups of polyhedra in terms of those of constituent sub-polyhedra.

Theorem 3.5.13 (Van Kampen) Let M be a polyhedron with a triangulation J [ K where J; K; J \ K are all path-connected with inclusions jJ j j - jJ \ K j ,!i jK j : Then, for all vertices v in J \ K ,

1 (jM j; v)  = 1 (jJ j; v)  1 (jK j; v)=

where  denotes the set of relations:

j (z ) = k (z )

for all z 2 1 (jJ \ K j; v) :

Proof: The trick is to take a maximal tree T0 in jJ \ K j and extend it to give maximal trees T1 ; T2 in J; K , respectively. Then T1 [ T2 is a maximal tree in J [ K . Next construct 1 (jM j; v) as G(M; T1 [ T2 ). Finally, the result follows because j (1 (jJ \ K j; v)) is generated by edges of J \ K n T0 in J; and k (1 (jJ \ K j; v)) is generated by edges of J \ K n T0 in K:



Ex on Van Kampen's theorem 1. Use Van Kampen's theorem to nd 1 (X ) for the following spaces: (a) S1 _ S1 _ S1; (b) RP 2 , as the union of a 2-disk and a Mobius strip; (c) A Klein bottle as a union of two Mobius strips. 2. Every nitely presented group G admits a representation as the fundamental group of some simplicial complex K .

3.6. SIMPLICIAL AND CELLULAR APPROXIMATION

89

3.6 Simplicial and cellular approximation The value of performing barycentric subdivisions repeatedly for a given simplicial complex K lies in the property that, up to homotopy, any continuous map between simplicial complexes can be replaced by a simplicial map on a suitable subdivision. This is the Simplicial Approximation Theorem (cf. Gray [38] x12 for a full discussion). In fact, we use a closely related result that is more directly suited to our program for exploiting the greater generality oered by CW-complexes. We show that, for a CW-complex X , any map f : Sn ! X is equivalent, up to homotopy, to a map that is linear where it crosses the interior of cells in X (cf. page 77 and Switzer [106], p. 76 et seq.).

Theorem 3.6.1 (Simplicial approximation (SAT)) Assume (X; A) is a relative CW-complex with X = A [ en and let K be a nite simplicial complex with a subcomplex L. Then, given a map of pairs f : (jK j; jLj) ! (X; A) ; there exists a subdivision (K 0 ; L0) of (K; L) and a map f 0 satisfying: (i) f and f 0 agree on f A; (ii) f  f 0 (rel f A);

(iii) if f 0 (jj) meets f(x) j kxk  1=4; x 2 B n g where  is the characteristic map of en and  2 K 0 , then f 0 (jj)  e0 , and f 0 is linear on jj.

Proof: Denote

en1=4 = f(x) j kxk  1=4; x 2 B n g and similarly use en1=2 ; en3=4 . Observe that jK j is compact so  1 f is uniformly continuous on  en3=4 . Subdivide K until no simplex of K 0 has diameter more than  where  satis es:

k 1 f (x)  1 f (y)k  1=4 whenever kx yk   for x; y 2  en3=4 :

This eects a trichotomy on K 0 , giving disjoint classes:

C1 = f 2 K 0 j f (jj)  X nen1=2g ; C2 = f 2 K 0 j f (jj)  X nen1=2g ; C3 = f 2 K 0 j f (jj) \ @en1=2 = ?g : Thus, we have arranged that those  2 C3 do not meet en1=4 when mapped by f and, moreover, neither does the convex hull of this image. We begin the construction of f 0 as follows:  If  2 C1 de ne f 0 = f on jj.

90

CHAPTER 3. HOMOTOPY GROUP THEORY

 If (v0 ; v1 ; : : : ; vk ) =  2 C2 and x = ri vi 2 jj then put:

f 0 (x) = ri f (vi ), so f 0 is constructed linearly from f .  For (v0 ; v1 ; : : : ; vk ) =  2 C3 we use induction on dim  = k. If k = 0; jj = v0 and we put f 0(v0 ) = f (v0 ).  Next suppose that we have de ned f 0 on all  with dim  < k such that f 0 (jj) is contained in the convex hull of f (jj). Then given  = (v0 ; v1 ; : : : ; vk ) we already have f 0 de ned on j@j and that image lies inside the convex hull of f (j@j). 1 ; i = 0; 1; : : : ; k . Now,  The barycenter b of  has coordinates ri = (k+1) each x 2 jjnfb g has a unique expression in the following form: x = tb + (1 t)yx : for some t 2 [0; 1] and some yx 2 j@j. Since f 0 (yx) is already well de ned we put f 0 (x) = tf (b ) + (1 t)f (yx) :

This linear combination ensures continuity on jj and traps f 0 (jj) inside the convex hull of f (jj). Thus f 0 is well de ned and meets conditions (i) and (iii). A homotopy to satisfy (ii) is

 f (x)

H : jK j  I ! X : (x; t) 7 ! (1 t)f (x) + tf 0(x) ifif xx 22  22= CC1 ;: 1 This is evidently continuous, and stationarity on C1 ensures that the homotopy is relative to f A. 

Ex on simplicial approximation 1. Prove that the set of xed points of a simplicial map between polyhedra

s : jK j ! jLj is a subcomplex of the rst barycentric subdivision of K but not necessarily of K itself. 2. Consider simplicial approximation to the map

f : [0; 1] ! [0; 1] : x 7 ! x2 between standard 1-simplices to show how barycentric subdivision of the domain helps. 3. Use the simplicial approximation theorem to show that the set of homotopy classes of continuous maps, between two compact polyhedra, is at most countable.

3.6. SIMPLICIAL AND CELLULAR APPROXIMATION

91

4. Consider a simplicial loop

s : j@ 42 j ! j@ 4n+1 j ; where 4m is the standard m-simplex. If n  2, then s is inessential and so, by the simplicial approximation theorem, Sn is simply connected. 5. Let K be a simplicial complex of dimension k < n. Then a simplicial map s : jK j ! j@ 4n+1 j is inessential; so by the simplicial approximation theorem we have k (Sn) = 0 for all k < n. 6. Simplicial maps are dense in each space of continuous maps between polyhedra.

Theorem 3.6.2 (The property of being n-connected) If (X; A) is a relative CW-complex and (X; A)n is obtained from (X; A)n 1 by attaching n-cells, with A = (X; A) 1 , then for each n  0:  every path component of X meets (X; A)n ;  r (X; (X; A)n ; y) = 0 for r = 1; 2; : : :; n and all y 2 (X; A)n . We call (X; (X; A)n ) n-connected if both properties hold.

Proof: We know that (X; (X; A)n ) has both of the properties if and only if, for all f : (B r ; Sr 1) ! (X; (X; A)n ) and all r  n, f  f 1(rel Sr 1) for some f 1 with f 1 B r  (X; A)n . We proceed as follows for each r  0. Given such a map f , then f B r is compact and therefore contained in (X; A)m m m for some m; moreover, f B r meets only nitely many m-cells, em 1 ; e2 ; : : : ; eq . We

shall deal with these cells in turn. Let m > n and apply simplicial approximation to get f 1 , homotopic to f relm 1 r em1 is at most ative to f ((X; A)m nem 1 ) and linear on (e1 )1=4 . Since (f B ) \  m 1 r 1 r-dimensional, and r  n < m, we can nd x 2 e nf B and a deformation retraction

D : ( (X; A)m nfxg )  I ! (X; A)m nfxg onto (X; A)m nem1 : This allows a deformation of f 1 by means of a homotopy rel f ((X; A)m nem 1 ), given by H : B r  I ! X : (y; t) 7 ! D(f 1 (y); t) into H ( ; 1) = f 2 with f 2 B r  (X; A)m nem 1. Evidently we can repeat this process until some deformation f m is found with

f m B r  ( (X; A)m n [2j=1 emj ) = (X; A)m

1

CHAPTER 3. HOMOTOPY GROUP THEORY

92

and, by transitivity of homotopy, f  f m (rel Sr 1). By means of this we can, up to homotopy (rel @ B r ), exclude f B r from all of the m-cells and indeed from each m-skeleton for m > n until eventually we nd some f^ with f  f^(rel @ B r ) and f^B r  (X; A)n : Since f was arbitrary, r (X; (X; A)n ) = 0, and the base point is irrelevant because X is path connected to (X; A). 

Corollary 3.6.3 Given a CW-complex X and the inclusion X n ,!i X of its nskeleton for n > 0, then

i : r (X n) ! r (X ) is

 an isomorphism for r < n

an epimorphism for r = n:

Proof: Substitution of r (X; X n) = 0 for r  n in the exact homotopy sequence for the pair gives

  (X ) ! 0 0 ! r (X n ) i! r

(r < n)

  (X ) ! 0 (r = n) : n+1 (X; X n) ! n (X n ) i! n These yield the required properties for i



Corollary 3.6.4 r (Sn) = 0 for r < n Proof: Sn is a CW-complex with cells e0 and en so its (n 1) skeleton is just the basepoint e0 and hence r ((Sn)m 1 ) = 0 for all r. Now, when we again use the exact sequence to yield a surjection

i : r ((Sn)m 1 ) ! r (Sn); for all r  m 1 : It follows that r (Sn) = 0 for r < n.



We are now in a position to prove the cellular version of the simplicial approximation theorem, in a conveniently general form for pairs (cf. Spanier [97], p. 404 et seq.).

Theorem 3.6.5 (Cellular approximation) Given f : (X; A) ! (Y; B) in Topp between relative CW-complexes, then the extension problem in CW-pairs

(K; C ) (Y; B ) has a solution y  f (rel K ).

(X; A)

vv vv v v vv y vv

 

/

{

3.7. WEAK HOMOTOPY EQUIVALENCE IS GOOD ENOUGH IN CW

93

Proof: We have seen that if (X; A) is a relative CW-complex then for any k  0; (X; (X; A)k ) is k-connected. Moreover, the inclusion A ,! X is a co bration.

Hence by induction on the skeletons of (X; A) we obtain a sequence fHk : (X; A)  I ! (Y; B ) j k  0g of homotopies relative to (X; A)k 1 which yields a stepwise deformation of f into a cellular map (we may suppose that it is already cellular on K):

f H0 f0 H1 f1 


 fk 1 Hk fk 


;

with

fk (X; A)k  (Y; B )k : H y with Then the required y is obtained by a homotopy f  H : (X; A)  I ! (Y; B ) ! t (1 k1 ) : (x; t) 7 ! Hk 1 x; 1 1 k

k+1

1 ]. for t 2 [1 k1 ; 1 k+1  Corollary 3.6.6 Any continuous map between relative CW-complexes can be replaced by a cellular map, up to homotopy, for we can take K to be a 0-cell in X .  The reason that cellular approximation works is that given two CW-complexes (X; A) and (Y; B ), with (Y; B ) being n-connected for all n, we can, up to homotopy (rel A), replace any continuous f : (X; A) ! (Y; B ) by a cellular map f^ : (X; A) ! (Y; B ). Indeed, if X nA has no cells of dimension greater than m, then we can nd such f^  f (rel A) provided that (Y; B ) is m-connected. The trick is to climb the relative skeletons, exploiting their n-connectedness to construct the required sequence of homotopies. The mole's living space is 3-dimensional but its 2-skeleton is of interest to the worm.

3.7 Weak homotopy equivalence is good enough in CW

Our next objective is the J. H. C. Whitehead theorem, which almost says that two CW-complexes are homotopic if and only if they have the same homotopy groups (and the same number of path components). De nition 3.7.1 A map f : X ! Y in Top is called an n-equivalence if (for some n  1 ) the induced maps fr (X; x) ! r (Y; f (x)) (8 x 2 X ) are isomorphisms for 0  r < n and an epimorphism for r = n. If f is an nequivalence for all n  1, then we call f a weak homotopy equivalence or an 1-equivalence .

CHAPTER 3. HOMOTOPY GROUP THEORY

94

It follows immediately that n-equivalences are preserved by compositions of maps and by homotopic deformations. Hence, in order for f : X ! Y to be an nequivalence, it is necessary and sucient that the inclusion of X in the mapping cylinder Mf of f be an n-equivalence, because we have Mf  Y as a deformation retraction (cf. p. 23). The exact homotopy sequence for (Mf ; X ) yields (for n  0)   (Y ) !  (M ; X ) : n+1 (Mf ; X ) ! n (X ) f! n n f

So f is an n-equivalence precisely when (Mf ; X ) is n-connected. Note the conversion of the problem of deciding when f is an isomorphism. It becomes the problem of the vanishing of certain homotopy groups in an exact sequence containing f .

Theorem 3.7.2 (Extension for n-equivalence) Let (P; Q) be a relative CWcomplex such that P n Q has no cells of dimension higher than n and let f : X ! Y be an n-equivalence. Then there is a solution gy to the extension problem in the following diagram; namely, gy q = g and fgy  h (rel Q). q

Q

P

w y www g w g h w ww w w X4 f Y 44



44

44



/





{

/

h

Mf 



Proof: Let f : Mf ! Y be a homotopy inverse of j . We can construct a H jhq such that rH is constant. Next, (P; Q) is a relative CWhomotopy ig  complex so q is a co bration and we can nd h0 : P ! Mf such that h0 q = ig and rh0  rjh(rel Q). Hence h0 (Q)  (X  f0g)  = X , so we can consider h0 as a map of pairs (P; Q) ! (Mf ; X ). But (Mf ; X ) is n-connected because f is an n-equivalence; moreover, P n Q has no cell of higher dimension than n. Thus h0 is homotopic (rel Q) to some gy : P ! X . Finally, gy q = g and fgy = rigy  rh0 (rel Q); h = rjh  rh0 (rel Q):



Corollary 3.7.3 Given an n-equivalence f : X ! Y and a CW-complex P having no cells of dimension above n, then the induced map f [P; X ] ! [P; Y ] is epic; and if P has no cells of dimension n, then it is also monic.

Proof: The surjectivity follows from the previous theorem with Q = ?, for given

[h] we nd [gy ]. To show that f is injective if dim P < n, consider the relative

3.7. WEAK HOMOTOPY EQUIVALENCE IS GOOD ENOUGH IN CW

95

CW-complex (P  I; P  @ I) where @ I = f0; 1g. Suppose we have g0  g1 with fg0  fg1 . Then we can nd g : P  @ I ! X satisfying (8 a 2 P ) g0 (a) = g(a; 0) g1 (a) = g(a; 1) and h : P  I ! with hq = fg. P  @I q P  I /

g

P

g0 g1

h

/

/

X 

Y 

f /

Now, dim(P  I)  n, so we reapply the theorem to nd gy q = g. But this gy is itself a homotopy from g0 to g1 so [g0 ] = [g1 ] and f is injective.  The next deduction is an important theorem. Observe that the isomorphism of homotopy groups must be induced by a map in order to ensure a homotopy equivalence. Switzer [106, p. 90] gives an example (the lens spaces) where this necessity is illustrated. Corollary 3.7.4 (J. H. C. Whitehead weak homotopy theorem) A map between CW-complexes is a homotopy equivalence if and only if it is a weak homotopy equivalence.

Proof: ()) If f : X ! Y is a homotopy equivalence, then it induces bijections on

sets of homotopy classes and hence it is a weak homotopy equivalence. (() If f : X ! Y is a weak homotopy equivalence between CW-complexes, then by Corollary 3.7.3, we have induced bijections: [X; X ] ! [X; Y ] and [Y; X ] ! [Y; Y ] : Take any g : Y ! X with f[g] = [1Y ], so fg  1Y . We see that gf  1X as follows, f [gf ] = [fgf ] = [1Y f ] = [f 1X ] = f [1X ] ; because f is injective. 

Ex It is important to note that the Whitehead theorem applies only to CWcomplexes. Maunder [68, p. 301] gives a detailed study of the subspace X  R2 given by the set of all line segments: from (0; 1) to both (0; 0) and (1=n; 0) and from (0; 1) to both (0; 0) and ( 1=n; 0), for natural n. This space is not contractible but n (X ) = 0 for all n  0. Hence f : X ! X : (x; y) 7 ! (0; 0) is a weak homotopy equivalence but not a homotopy equivalence, because X is not a CW-complex.

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CHAPTER 3. HOMOTOPY GROUP THEORY

3.8 Exploiting n-connectedness We now prepare for the excision theorem. Once again, in order to nd when a certain map induces isomorphisms, we embed the induced homomorphism into an exact sequence and seek the conditions for appropriate homotopy groups to vanish. The map in question here is

j : (A; A \ B ) ,! (A [ B; B ) and we shall see that a convenient exact homotopy sequence in which to embed j is that of the (pointed) path space pair

P (A; A \ B ) ,! P (A [ B; B ) : First recall that in this notation the relevant de nitions are

n (A [ B; B ) = n 1 (P (A [ B; B )) = 0 n 1 (P (A [ B; B )) n (A; A \ B ) = n 1 (P (A; A \ B )) : The exact sequence is then, for n  2, n (P (A [ B; B ); P (A; A \ B )) ! n (A; A \ B ) ! n (A [ B; B ) ! n 1 (P (A [ B; B ); P (A; A \ B )) : Hence from the properties of such sequences, we deduce that j is: 9 isic for 2  r < n = epic for r = n ; () r 1 (P (A [ B; B ); P (A; A \ B )) = 0 monic for r = 1 for 2  r  n. Theorem 3.8.1 (Homotopy excision) Let X = A [ B be a pointed space with basepoint in A \ B . Consider the inclusion j : (A; A \ B ) ,! (A [ B; B ). For n  1, suppose that (A; A \ B ) is an n-connected CW-complex and (B; A \ B ) is an m-connected CW-complex. Then j : r (A; A \ B ) ! r (A [ B; B ) is isic for 1  r < m + n and epic for r = m + n.

Proof: We merely indicate the steps, full details are given in Switzer [106, pp. 81{ 84]. The procedure depends on how A \ B needs to be augmented to construct A and B . 1. If A = A \ B [ en1 and B = A \ B [ em1 , for n1 > n and m1 > m, then r 1 (P (A [ B; B ); P (A; A \ B )) = 0 for 2  r  n1 + m1 2 :

3.8. EXPLOITING N -CONNECTEDNESS

97

To establish this, we represent each element of the homotopy set on the left by the homotopy class of a map of quadruples

f : (B r 1  I; @B r 1  I; @B r 1  I; B r 1  f0g [ fg  I)

! (A [ B; A; B; ) ;

which is possible because P (A [ B; B ) consists of pointed paths in A [ B ending in B , and also r 1 r 1 ((A [ B )I)B  = (A [ B )B I: Then the construction uses simplicial approximation on B r 1  I to achieve a suciently ne triangulation relative to en1 and em1 for the given f to be deformed homotopically into a map that is nullhomotopic. 2. If A = A \ B [ en1 [ en2 [


[ enk and B = A \ B [ em1 with m1 > m and ni > n for 1  i  k, then we apply (1) repeatedly and use induction. 3. The method of (2) extends also to the case

B = A \ B [ em1 [ em2 [


[ eml : 4. Given (A; A \ B )n = A \ B = (B; A \ B )m and f as in (1), then we can nd A1 ; B 1 with

A \ B  A1  A; A \ B  B 1  B

and A1 ; B 1 representable in the form of case (3), and moreover such that the (compact) image of f lies in A1 [ B 1 . Then it follows from (3) that

r 1 (P (A1 [ B 1 ; B 1 ); P (A1 ; A1 \ B 1 )) = 0 for 2  r  m + n : 5. If (B; A \ B )m = A \ B , then we can construct (A1 ; C 1 ) with n-skeleton C 1 and with A \ B as a strong deformation retract of C 1 and A a strong deformation retract of A1 . It follows that A \ B is a strong deformation retract of A1 [ B and B is a strong deformation retract of C 1 [ B = B 1 . Hence the inclusion (A [ B; A; B ) ,! (A1 [ B; A1 ; B 1 ) induces the following isomorphisms for r  0:

r 1 (P (A [ B; B ); P (A; A \ B ))  = r 1 (P (A1 [ B; B 1 ); P (A1 ; A1 \ B 1 )) :

However, by (4), the group on the right is trivial for 2  r  m + n. 6. The procedure of (5) can be used in the general case by embedding (B; A \ B ) in (B 1 ; C 1 ) to induce isomorphisms by: (A [ B; A; B ) ,! (A [ B 1 ; A [ C 1 ; B 1 ) :



CHAPTER 3. HOMOTOPY GROUP THEORY

98

Corollary 3.8.2 (Relative to quotient homotopy homomorphisms) Given a relative CW-complex (X; Y ) with Y m-connected and (X; Y ) n-connected, then the projection onto the quotient p : (X; Y ) ! (X=Y; fg) induces a map

p : r (X; Y ) ! r (X=Y; fg) which is isic for 2  r  m + n and epic for r = m + n + 1.

Proof: We embed (X; Y ) in (X [ CY; CY ) and use the homeomorph (X [ CY=CY; )  = (X=Y; ) ; observing that CY ' . Now, we can apply the excision theorem to the case  r 1 (Y ) = 0 for A = X; B = CY because (X; Y ) is n-connected and r (CY; Y ) = 1  r  m + 1. Hence we have a homomorphism. i : r (X; Y ) ! r (X [ CY; CY ) which is isic for 2  r  m + n and epic for r = m + n + 1. The result follows because (X [ CY; CY ) ' (X [ CY=CY; fg).  Theorem 3.8.3 (Freudenthal suspension theorem) If X is a CW-complex which is n-connected for some n  0, then the suspension functor S induces a homomorphism

S : r (X ) ! r+1 (SX ) : [f ] 7 ! [sf ] which is isic for 1  r  2n and epic for r = 2n + 1.

Proof: First recall that SX = S1 ^ X and Sf = 1S1 ^ f ; hence S (Sr) = S1 ^ Sr  = Sr+1, and so S is well de ned. Next, CX=X  = SX and n+1 (CX=X ) = n+1 (CX=X; fg). Now X is n-connected and (CX; X ) is (n + 1)-connected so, by

the foregoing Corollary,

p : (CX; X ) ! (CX=X; fg)  = (SX; fg)

induces p which is isic for 2  r  2n + 1 and epic for r = 2n + 2. The result follows by virtue of the isomorphisms

r (X )  = r+1 (CX; X )  r+1 (SX ) = r+1 (CX=X )

of homotopy groups.

Corollary 3.8.4 (Spheres hold water) n (Sn) = Z for n  1.



3.9. EXTRACTING HOMOTOPY GROUPS FROM KNOWN BUNDLES

99

Proof: We have already seen that 1 (S1) = Z. Next, the Hopf bration S3 ! S2 with ber S1 gives an exact sequence containing 2 (S3) ! 2 (S2) ! 1 (S1) : But we have shown that r (Sn) = 0 for r < n so 2 (S2)  = 1 (S1)  = Z: Now we can apply our suspension theorem: Sn is (n 1)-connected hence S : r (Sn) ! r+1 (Sn+1) is isic for r  2(n 1) and so n (Sn)  = n+1 (Sn+1) for n  2 :  This is, of course, not the end of the story for spheres; for although we have shown that r (S1) = 0 for r > 1, in general r (Sn) 6= 0 for r > n. Indeed we actually encounter such a case elsewhere in the exact sequence for the bration used above: 3 (S2)  = Z.

3.9 Extracting homotopy groups from known bundles

p Given a bundle F ,! E ! B we can take advantage of any topological simplicity in the pair (E; p fg) to yield information about homotopy groups from the exact sequence. For example:

 If p fg ,!i E is homotopic to the constant map in E , so p fg is contractible in E , then for n  2 n (E; p fg)  = n (E )  n 1 (p fg) ; hence

n (B )  = n (E )  n 1 (p fg) :

 If r : E ! p fg is a retraction, then for n  2 n (E )  = n (E; p fg)  n (p fg) ; hence

n (E )  = n (B )  n (p fg) :

 If f : E ! p fg is homotopic (mod fg) to 1E , then for n  2 n (p fg)  = n (E )  n+1 (E; p fg) :

100 In

CHAPTER 3. HOMOTOPY GROUP THEORY

The Hopf sphere bundles allow us to deduce some homotopy groups of spheres.

p 2 S1 ,! S3 ! !S ; 3 3 1 (S ) = 0, so p fg is contractible in S . Then n (S2)  = n (S3)  n 1 (S1) for n  2 ;

and so

n (S2)  = n (S3) for n  3 :

Now we can deduce the result of Hopf that established homotopy theory: 3 (S2)  =Z p 4 In S3 ,! S7 ! S , p fg is contractible in S7 and so: n (S4)  = n (S7)  n 1 (S3) for n  2 : But we know that  0 ; n < k; k  n (S ) = Z ; n = k: This gives n (S4)  = n 1 (S3) for 2  n < 7 7 (S4)  = 6 (S3)  Z : Again, in

S7 ,! S15 !p S8 ;

p fg is contractible in S7 and so: n (S8)  = n (S15)  n 1 (S7) for n  2 : Hence

n (S8)  = n 1 (S7) for 2  n  15 15 (S8)  = 14 (S7)  Z :

Homotopy groups of spheres are tabulated in Appendix D. We can now compute some homotopy groups of classical Lie groups. In some cases we shall need the Stiefel varieties and Grassmann manifolds, so we review them rst. Let F denote R, C , or H , respectively. There are natural inclusions F n ,! F n+1 : x 7! (x; 0) so we can de ne F 1 = lim! F n and an induced inner product hji on it. De nition 3.9.1 The Stiefel variety of (orthonormal) k-frames in n-space is Vk (F n ) = f(x1 ; : : : ; xk ) 2 F nk j hxi jxj i = ij g and Vk (F 1 ) = lim! Vk (F n ). The Grassmann manifold or Grassmannian of k-planes in n-space is Gk (F n ) = fk-dimensional subspaces of F n g and Gk (F 1 ) = lim! Gk (F n ). When F is R or C , there are also oriented versions SGk (F n ) and SGk (F 1 ) in which the k-planes are oriented subspaces.

3.9. EXTRACTING HOMOTOPY GROUPS FROM KNOWN BUNDLES 101

Ex on Stiefel and Grassmann spaces 1. Vn (F n )  = O(n); U (n); Sp(n), respectively. 2. For k < n,

O(n) = SO(n) Vk (Rn )  = O(n k)  SO(n k) U (n) = SU (n) Vk (C n )  = U (n k)  SU (n k) Sp(n) Vk (H n )  = Sp(n k) V1 (F n )  = Sn 1; S2n 1; S4n 1, respectively. Vn 1 (Rn )  = SO(n) and Vn 1 (C n )  = SU (n). For 1  l < k  n, Vk (F n ) ! ! Vl (F n ) is a ber bundle. The natural projection Vk (F n ) ! ! Gk (F n ) sending a frame to its span de nes a principal O(k); U (k); Sp(k)-bundle, respectively (cf. Chapter 7). Similarly, Vk (Rn ) ! ! SGk (Rn ) is a principal SO(k)-bundle and Vk (C n ) ! ! SGk (C n ) is n n a principal SU (n)-bundle, and SGk (F ) is a cover of Gk (F ). 7. For k < n, O(n) O(n) SGk (Rn )  Gk (Rn )  = SO(k)  O(n k) = O(k)  O(n k) U (n) U (n) Gk (C n )  SGk (C n )  = U (k)  U (n k) = SU (k)  U (n k) Sp(n) Gk (H n )  = Sp(k)  Sp(n k)

3. 4. 5. 6.

8. We have G1 (F n )  = RP n 1 ; C P n 1 ; H P n 1 , respectively. Also, SG1 (Rn )  = n 1 n S and SG1 (C )  = S2n 1. 9. i (Vk (F n )) = 0 for i  n k 1; i  2(n k); i  4(n k) + 2, respectively. Thus i (Vk (F 1 )) = 0 for all i and k, whence Vk (F 1 ) is contractible. 10. The rst nontrivial homotopy group of Vk (F n ) is:

n k (Vk (Rn ))  = 2(n k)+1 (Vk (C n ))  = n  4(n k)+3 (Vk (H )) =

Z

for k = 1 or n k even,

Z2 for k  2 and n k odd;

Z; Z:

Homotopy groups of Stiefel varieties are also tabulated in Appendix D.

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CHAPTER 3. HOMOTOPY GROUP THEORY

Ex on homotopy groups of Lie groups 1. For every Lie group G, 2 (G) = 0. 2. For every topological group G, 0 (G) is an abelian group. In particular, 0 (O(n))  = Z2 and 0 (O(1; n))  = Z2  Z2, the Klein 4-group. 3. 0 (SO(n))  = 0 (U (n))  = 0 (SU (n))  = 0 (Sp(n))  = 0 (Spin(n)) = 0.

4. Since SO(n) is the identity component of O(n), all their homotopy groups after the zeroth coincide, so we'll not list O(n) any more. Thus 1 (SO(1))  = 1 (SU (n))  = 1 (Sp(n)) = 0 for n  1, 1 (SO(2))  = 1 (U (n))  = Z for n  1, and 1 (SO(n))  = Z2 for n  3.

5. k (SO(2))  = k (U (1)) = 0 for k  2. 6. 3 (U (n))  = 3 (SU (n))  = Z for n  2, 3 (Sp(n))  = Z for n  1, 3 (SO(3))  =   Z, 3 (SO(4)) = Z  Z, and 3 (SO(n)) = Z for n  5.

7. Use k+1 (Sk)  = Z2 for k  3 to calculate 4 (SO(n)) and 4 (Sp(n)) for n  4, and 4 (U (2)) and 4 (SU (2)). 8. From the inclusion O(n) ,! O(n + 1) we obtain ses

n even: 0 ! Z ! n 1 (O(n)) ! n 1 (O(n + 1)) ! 0 n odd: 0 ! Z2 ! n 1 (O(n)) ! n 1 (O(n + 1)) ! 0 9. Some of these groups are stable in the following sense. Considering the bration Sn  = SO(n + 1)=SO(n) 2 n S +1  = SU (n + 1)=SU (n) 2 n S +1  = U (n + 1)=U (n) 4 n +3  S = Sp(n + 1)=Sp(n)

yields k (SO(n))  = k (SO) for k  n 2; k (SU (n))  = k (SU ) for k  2n 1;  k (U (n)) = k (U ) for k  2n 1; k (Sp(n))  = k (Sp) for k  4n + 1:

Here, O = lim! O(n), SO = lim! SO(n), U = lim! U (n), and Sp = lim! Sp(n) are the in nite or stable groups. 10. For the spinor groups of low rank, the identi cations in Appendix A together with the tables of Appendix D allow the computation of their homotopy groups through 23 . Since k (SO(n))  = k (Spin(n)) for k  2, we may also compute k (SO(n)) for k  23 and n  9. Since k (U (n))  = k (SU (n)) for k  2, we can also compute k (U (n)) for k  23 and n  4.

11. For n  12 and k  2n 1, k (SO(n))  = k (SO)k+1 (Vn (R2n ))  = k (SO) 1 n 1 k+1 (RP =RP ).

3.9. EXTRACTING HOMOTOPY GROUPS FROM KNOWN BUNDLES 103 Finally, recall that G2 and F4 are two of the exceptional simple Lie groups, and that the Cayley projective plane is O P 2  = F4 =Spin(9). The following berings may occasionally be found useful, in which a bration F ,! E ! ! B is denoted by B F ! E.

>

V2 (R7)

S3  = Sp(1) = S5

SU (2) 

SU (3) /

=

S6

G2 

/

S7

Spin(3)O 

SU (4)  = Spin(6)

S7



OOO OOO OO S7 OOOO O

O

/

S15 /

Spin(9) OP 2

S5

Sp(2)  = Spin(5) '

Spin(7) 

S6

F4 

Problem Find a nice, `geometric' visualization of the connecting map @ in the exact homotopy sequence of a ber bundle (or of a bration, if you can).

Chapter 4

Homology and Cohomology Theories `Well, if you knows of a better 'ole, go to it.'|Bairnsfather

4.1 Introduction We ended the last chapter by calculating homotopy groups for spheres, which in view of their importance in CW-complexes is a necessary starting point for our program to classify spaces up to homotopy. However, in general the homotopy groups are dicult to compute so we turn to a coarser theory, homology, which yields more easily computed groups but by design has strong ties with homotopy theory. There are many excellent books which give leisurely accounts of homology theory, gradually revealing its versatility through particular representations. We wish to use the theory and from the outset we know very well certain properties that will be convenient to have. Accordingly, we shall declare these properties as axioms, see why they are chosen, and then show that they can be realized in a variety of ways. At each stage in the development of homology theory there is a dual situation and the corresponding theory is called cohomology; it arises from cofunctors whereas homology arises from functors. In either case we shall want the theory to yield diagrams in the category Ab of abelian groups. However, depending on the circumstances, we shall want to use the theory on several topological categories:

   

T] opp = topological pairs and homotopy classes of pair maps; ^ = CW-complex pairs and homotopy classes of CW-pair maps; CWW g = pointed spaces and homotopy classes of pointed maps; Top g  = pointed CW-complexes and homotopy classes of pointed CW-maps. CW 105

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

106

Observe that already we are planning to ignore changes in maps up to homotopy. Since we have the inclusions

g ; g ,! Top ^ ,! T] CWW opp and CW 

the main dierences among our domain categories are between pairs and pointedness. There are two important internal functors corresponding to these dierences:

Restriction

(A; ?)

(X; A)

R : T] opp /

T] opp : [f ] 

(Y; B )

(B; ?)

X

SX



Suspension

g S : Top /

[f jA ] /

g : [f ] Top 



[Sf ] /

Y 

SY 

We shall want our theories to behave well with respect to these functors. From our experience with homotopy groups we are prompted to have excision inclusions yielding isomorphisms and pair inclusions yielding exact sequences. These points summarize our requirements for homology and cohomology: (i) (ii) (iii) (iv)

functoriality; naturality with respect to restriction R or suspension S ; exactness for inclusions; isomorphisms for excisions.

There are then, a priori, two families of theories: one on T] opp using (i), (ii) R, (iii) and (iv); g using (i), (ii) S and (iii). one on Top These items are similarly labelled in the following de nitions.

4.2 Homology and cohomology theories Homology Axioms g consists of four parts: A homology theory H on Topp

4.2. HOMOLOGY AND COHOMOLOGY THEORIES

107

(i) a sequence fHn j n 2 Zg of functors Hn : T] opp ! Ab; (ii) natural transformations @n : Hn 7! Hn 1  R; [i ] [j ] (iii) the inclusions (A; ?) ,! (X; ?) and (X; ?) ,! (X; A) induce exactness for all n 2 Z in

Hn (A) Hn [i] Hn (X ) Hn [j] Hn (X; A) /

@n

/

/

Hn 1 (A)

where, as usual, Hn (X ) = Hn (X; ?); [k ] (iv) excision the inclusion (X nU; AnU ) ,! (X; A) with U  A, induces for all n 2 Z isomorphisms

Hn (X nU; AnU ) Hn [k] Hn (X; A) : /

g has three parts: A homology theory H~  on Top

g ! Ab; (i) a sequence fH~ n j n 2 Zg of functors H~ n : Top (ii) natural equivalences n : H~ n 7! H~ n+1  S ; [i ] [j ] (iii) A ,! X and X ,! X [ CA induce exactness for all n 2 Z in ~ ~ H~ n (A) Hn [i] H~ n (X ) Hn [j] H~ n (X [ CA) ; /

/

where CA = I ^ A is the cone on A.

Cohomology Axioms g is A cohomology theory H  on Topp (i) a sequence fH n j n 2 Zg of cofunctors H n : T] opp ! Ab; (ii) natural transformations n : H n  R ! H n+1; [i ] [j ] (iii) inclusions (A; ?) ,! (X; ?) and (X; ?) ,! (X; A) induce exactness for all n 2 Z in

H n+1 (X; A) o

n

n n H n (A) H [i] H n (X ) H [j] H n (X; A) ; o

o

(iv) excision inclusion (X nU; AnU ) ,[!k] (X; A) with U  A induces isomorphisms for all n 2 Z n H n (X nU; AnU ) H [k] H n (X; A) : o

108

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

g is A cohomology theory H~  on Top (i) a sequence fH~ n j n 2 Zg of cofunctors H~ n : T] op ! Ab; (ii) natural equivalences n : H~ n 7! H~ n+1  S ; [i ] [j ] (iii) inclusions A ,! X and X ,! X [ CA induce exactness for all n 2 Z in ~n ~n H~ n (A) H [i] H~ n(X ) H [j] H~ n (X [ CA) ; o

o

where CA = I ^ A.

Sometimes homology and cohomology theories on T] op are called reduced theories; those on T] opp are then called unreduced theories. The de nitions could have been given on Top and Topp; then invariance up to homotopy would have been added as an axiom. In some texts (cf. Spanier [97], Gray [38]) the homology and cohomology theories are de ned by means of functors and cofunctors taking values in the category of graded abelian groups. There the objects are integer indexed sequences of abelian groups so the corresponding functors are fHn g; fH ng, and so forth. Elements of (co)homology groups are traditionally referred to as (co)homology classes As will be revealed in due course, homology and cohomology theories do indeed exist. The most important ones for practical purposes are ordinary (co)homology theories with coecients G. These have

H0 () = G and Hn () = 0 for n 6= 0 ; where  is any singleton, and a similar statement for cohomology. This is sometimes called the dimension axiom. The prototype candidate for the coecient group G is the in nite cyclic group Z and the universal coecient theorems allow us to obtain information on other coecient choices from this one. Under quite reasonable conditions we shall see eventually that any two such theories are naturally equivalent. We shall rst investigate the immediate consequences of our axioms. When there is no likelihood of confusion we shall denote Hn [f ] and H n [f ] by f and f , respectively.

4.3 Deductions from the axioms First of all there is a natural inclusion (X; x0 )

g Top

o



/

T] opp :



[f ]

(Y; y0 ) 

(X; fx0 g) [f ] /

(Y; fy0g) 

4.3. DEDUCTIONS FROM THE AXIOMS

109

We shall see how this allows us to construct a reduced theory from an unreduced theory, so our a priori dierences are less than might be expected. Consequently we give explicit instructions only for a homology theory on T] opp; passing to cofunctors will give the dual results for cohomology.

Functoriality

(fg) = f g ; (f 1 ) = (f ) 1 ; 1 = 1 : If f is a homotopy equivalence, then f is an isomorphism.

Inclusions Consider these four special cases of

A

o



i

X

/

~ ~~ ~ 1X 1A ~~ r ~ ~

A 

O



o



i

/

X

1. Weak retract: ri  1A 2. Retract: ri = 1A 3. Weak deformation retract: ri  1A; ir  1X 4. Deformation retract: ri = 1A; ir  1X . Since our homology functors are insensitive to dierences between  and =, there are only two cases: 1,2 and 3,4. We deduce the following.

Theorem 4.3.1 (Retracts split in homology) If A ,!i X is a (weak) retract, then for all n 2 Z Hn (X )  = Hn (A)  Hn (X; A) : Proof: We have a (weak) retraction r : X ! A so, in homology, (ri) = r i = 1 and for all n 2 Z there exists an exact sequence  H (X; A) @! H (A) i!  H (X ) j! 





@! Hn (A) i! n n n 1 j with (X; ?) ,! (X; A). But i is monic, so @ is trivial and j is an epimorphism

with a right inverse. Hence we have a split short exact sequence and the result follows, with ker r  = Hn (X; A) :  Corollary 4.3.2 (Deformation retracts kill relative homology) For a weak deformation retract A we have, for all n 2 Z, i r = 1 and so Hn (X; A) = 0. In particular, Hn (X; X ) = 0 and if (X; fx0 g) is contractible then Hn (X; fx0 g) = 0 for all n 2 Z. 

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

110

When the basepoint is well behaved, we have the following nice aid to computation.

Theorem 4.3.3 (Well-pointed spaces) If  ,!i X is a co bration, then Hn (X )  = Hn (X; )  Hn () : Proof: Being a co bration implies there exists p : X !  such that pi = 1, whence


! Hn () ! Hn (X ) ! Hn (X; ) !


splits for every n.



Ex If X is also path-connected, then H0(X ; G) = G for any ordinary H with coecients G.

Exactness Axiom (iii) gives us an exact homology sequence for a pair (X; A) and we can deduce the following.

Theorem 4.3.4 (Exact homology sequence of triple) If the triple (X; A; B)

has inclusions

c a b (A; ?) ,! (A; B ) ,! (X; B ) ,! (X; A) ; then there is an exact sequence in homology for all n 2 Z, @ H (A; B ) :  H (X; A) c!  H (X; B ) b! Hn (A; B ) a! n n 1 n

The composition c @ is sometimes denoted by
; it is a connecting morphism. Since the exact sequence can in principle be extended inde nitely, it is sometimes called long.

Proof: This is directly analogous to the situation for homotopy groups and involves a check of exactness at each group. A detailed checklist is provided in Switzer [106], pp. 42{43.  Corollary 4.3.5 (Cone-triple isomorphisms) In the triple (CA; A; fag), the pair (CA; fag) is contractible so we have an isomorphism for all n 2 Z : Hn (CA; A)  = Hn 1 (A; fag) :  Corollary 4.3.6 (Union-triple isomorphisms) For a CW-triad (A [ B; A; B), the inclusion k : (A; A \ B ) ,! (A [ B; B ) induces isomorphisms in homology.

4.3. DEDUCTIONS FROM THE AXIOMS

111

Proof: The trick is to construct a homotopy inverse to the inclusion i : (A; A [ B ) ,! (NA ; NA \ B ) where NA is an open neighborhood of A. Now, by the proposition there is an isomorphism j : Hn (NA ; NA \ B )  = Hn (A [ B; B ): This is so because (A [ B ) n A is open in B and NA is open, therefore A [ B = NA [ (A [ B n A)  NA [ B; where B denotes the interior of B . It follows that k is also an isomorphism. Details of the homotopy construction are given in Switzer [106], pp. 100{101.  Another kind of long exact sequence arises in ordinary homology theories when we begin with a ses of coecients 0 ! R ! N ! G ! 0. The connecting morphism in this case is called the Bockstein associated with the ses of coecients and is denoted by . Thus we have

Hn (X ; R) ! Hn (X ; N ) ! Hn (X ; G) ! Hn 1 (X ; R) ; and similarly for pairs (X; A).

Excisions There is a convenient equivalent form for the excision axiom which simpli es its application on CW-complexes.

g ! Ab satis es the Theorem 4.3.7 (Excision equivalent) A functor Hn : Topp excision axiom (iv) if and only if, for all triads (A [ B; A; B ) with A [ B = A [ B,

the inclusion

j : (A; A \ B ) ,! (A [ B; B ) induces isomorphisms; that is, for all n 2 Z :

Hn [j ] : Hn (A; A \ B )  = Hn (A [ B; B ):

Proof: Given axiom (iv), we can apply it to X = A [ B, with U = X n A. For, U = X n A so U  B. Also B n (X n A) = A \ B , hence j induces isomorphisms. Conversely, given the stated properties for a functor Hn , suppose that U  A  X and U  A. Then the triad ((X n U ) [ A; X n U; A) has (X n U ) [ A = (X n A) [ A since X = (X n A) [ A  (X n U ) [ A = (X n U ) [ A. Hence the inclusion (X n U; A n U ) = (X n U; (X n U ) \ A) ,! ((X n U ) [ A; A) induces the isomorphisms required by the excision axiom.



112

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

We can exploit the excision property to show that co bration projections induce isomorphisms in homology, a considerable simpli cation from the situation in homotopy theory; cf. Section 3.8.

Theorem 4.3.8 (Co brations induce homology isomorphisms) If A ,!i X

is a co bration then the projection p : (X; A) ! (X=A; fag) induces isomorphisms in homology.

Proof: Since i is a co bration, its mapping cone Ci (cf. Figure 4.1) is homotopy j equivalent to X=A. Now, Ci = X [ (CA) and so we have a triad inclusion (X; A) ,! (Ci ; CA) which induces isomorphisms in homology. But (CA; fag) is contractible

and Ci =CA is a homeomorph of X=A, so Hn (X; A)  = Hn (Ci ; CA)  = Hn (X=A; fag) :

A Ci

a





X

CA

i Figure 4.1: Mapping cone of inclusion A ,! X

Corollary 4.3.9 (Suspension isomorphism) If (X; x0) is a pointed space, then for all n 2 Z there is an isomorphism ~n : Hn (X; fx0 g) ! Hn+1 (SX; fg) : Proof: We have already found isomorphisms Hn+1 (CX; X )  = Hn (X; fx0 g) ; so we need only compose these with those induced by the homeomorphism CX=X  = SX and by the projection (CX; X ) ! (CX=X; fg).  Evidently, the ~n will be relevant to the construction of a reduced homology theory g where we require such a relationship between the homology and suspension on Top functors.

4.3. DEDUCTIONS FROM THE AXIOMS

113

Suspension Suspension is particularly important for spheres and we easily nd the following results for them, in any homology theory.

Theorem 4.3.10 (Periodicity in homology of spheres) For all m; n 2 Z with m  0 there are isomorphisms Hn (Sm; fg)  = Hn+1 (Sm+1; fg)  = Hn m (S0; fg)  = Hn m () : Proof: The rst two isomorphisms arise from suspensions Sm+1 = S (Sm) = S m+1(S0) The other one arises from the triad (S0; f 1g; f+1g) and inclusion (f 1g; ?) ,! (S0; f+1g).  We may now compute something interesting.

Corollary 4.3.11 (Ordinary homology of spheres) If H is ordinary with coecients G, then

8 G  G ; n = m = 0; < Hn (Sm)  = : G ; n = m 6= 0 or n = 0; m 6= 0; 0 ; otherwise.

Proof: From Theorem 4.3.3 we obtain Hn(Sm) = Hn(Sm; )  Hn() = Hn(Sm; ) for n = 6 0. From the preceding theorem we nd Hn (Sm; ) = Hn m (), whence



n = m; Hn (Sm; )  = G0 ;; otherwise: Combining these yields H0 (S0)  = G  G, and for m 6= 0, H0 (Sm)  = Hm (Sm)  = G.



This last result shows that if n 6= m, then the spheres Sn and Sm have dierent homotopy type and hence cannot be homeomorphic. On the other hand, Rn and Rm are both contractible so they have trivial reduced homotopy and the same homotopy type. However, by appropriately adding one point each to compactify Rn and Rm we get Sn and Sm, which would have to be homeomorphic if Rn and Rm were.

Corollary 4.3.12 (Simple invariance of domain) If n 6= m, then Sn and Sm, 

whence Rn and Rm , are not homeomorphic.

It is nontrivial to prove this in any other way!

114

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Ex Can you prove the full invariance of domain result now? If U1 and U2 are subsets of Sn, h : U1 ! U2 is a homeomorphism, and U1 is open, then so is U2 . (If

not, try it again after another chapter or so.)

We call a triad (A [ B; A; B ) excisive for a homology theory if the inclusion (A; A \ B ) ,! (A [ B; B ) induces isomorphisms in homology. Evidently we could equally well require the other inclusion, (A; A \ B ) ,! (A [ B; A) to induce isomorphisms; in fact they are equivalent conditions (cf. Switzer [106], pp. 103{4). From above, we know that CW-triads are always excisive so we shall be able to exploit their cellular structures to build homology information on their skeletons. A general and very useful consequence of excision is the following.

Theorem 4.3.13 (Mayer-Vietoris exact sequence) If (A[B; A; B) is excisive and C  A \ B , then for all n 2 Z there is an exact homology sequence  H (A [ B; C )
! H (A \ B; C ) :  H (A; C )  H (B; C ) ! Hn (A \ B; C ) ! n n n n 1

Proof: The candidate maps are clear enough from the inclusions. With (A; C )J

JJ JJ JJ

tttt 9

t tt

(A \ B; CJ )

(A [ B; C ) %

JJ J  JJJ

tt tt t tt  9

(B; C ) %

j

we have  = ( ;  ) and  (a; b) =  (a)  (b). Also, (A; A \ B ) ,! (A [ B; B ) k induces an isomorphism j by excision. Let (A [ B; C ) ,! (A [ B; B ) and (A \ l B; ?) ,! (A \ B; C ). Then we construct
as the composite 1 Hn (A [ B; C ) k Hn (A [ B; B ) j Hn (A; A \ B ) /

/

@

Hn 1 (A \ B ) l Hn 1 (A \ B; C ) : To establish exactness at Hn (A; C )  Hn (B; C ) we observe that  (a; b) = 0 =)  (a) =  (b) =) a; b 2 dom  so   (x) =  ( ;  )(x) =   (x)   (x) = 0 : 

/

4.3. DEDUCTIONS FROM THE AXIOMS

115

For the remainder, we use the two exact sequences generated by the triples (A; A \ B; C ) = (A; \; C ) and (A [ B; B; C ) = ([; B; C ) ; for the rows in the following commutative diagram:

Hn+1 (A; \) l@ =

/

Hn (\; C )  Hn (A; C ) i Hn (A; \) l@ Hn 1 (\; C ) /

/



Hn+1 ([; B ) 

=



Hn+1 (B; C )  Hn ([; C ) k Hn ([; B ) 

/

/



/



/

/

Hn (B; C )

By the four lemma, ker  =  ker  and  im  = im  . We deal with exactness at Hn ([; C ) and leave the other case as an exercise. Take (a; b) 2 dom  ; then
 (a; b) = l @j 1k ( (a)  (b)) but k  = 0 and  (a) 2 im  , so
 = 0 and im   ker
. Next, suppose that we have y 2 ker
. Then j 1 k (y) 2 ker l @ = im i and let j 1 k (y) = i(z ). But i (z ) = j 1 k  (z ) and k  (z ) = 0 since im   im  . Hence y 2 ker k = im  , so y =  (0; b) for some b. Therefore ker
 im  so ker
= im  and we have exactness at Hn ([; C ).



Ex Calculate H(Sn) for ordinary H using this theorem. Corollary 4.3.14 (Intersection isomorphisms) In particular, if C = A \ B

then  is an isomorphism, so also is

( ;  ) : Hn (A; A \ B )  Hn (B; A \ B )  = Hn (A [ B; A \ B ) 8 n 2 Z :



Corollary 4.3.15 (Disjoint union isomorphisms) For any disjoint open A; B, A [ B = A [ B so we have excision, and hence for all n 2 Z an isomorphism ( ;  ) : Hn (A)  Hn (B )  = Hn (A t B ) :  Corollary 4.3.16 (CW-triad isomorphisms) Any CW-triad is excisive; in particular for any pointed CW-complexes (A; a); (B; b), the triad (A _ B; A; B ) is excisive. Hence, for all n 2 Z we obtain an isomorphism ( ;  ) : Hn (A; fag)  Hn (B; fbg)  = Hn (A _ B; ) : 

116

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

4.3.1 Reduction and unreduction

g from Theorem 4.3.17 (Reduction) We can obtain a homology theory on Top g. one on Topp

Proof: Given (Hn ; @n) we de ne (H n ; n) by Hn (X; fx0 g)

(X; x0 )

g H n : Top /

Ab :



[f ]

Hn (Y; fy0 g)

(Y; y0 ) 

Hn (X; x0 ) n = ~n :

Hn [f ] /



= /

Hn (SX; ) =

=

Hn+1 (CX; X ) = Hn+1 (CX=X; ) 



/

The exactness axiom is satis ed because of isomorphisms H n (X [ CA; )  = Hn (X [ CA; fg)  = Hn (X [ CA; CA)  = Hn (X; A) which allow the exact sequence in H of the triple (X; A; fx0 g) to be reduced to the required one in H  .  The process of reduction is actually `reversible' if we require our reduced theory to satisfy the extra condition: weak homotopy equivalences induce isomorphisms, which is usually abbreviated to WHE. Theorem 4.3.18 (WHE unreduction) If (H~ n ; n) is a reduced theory satisfying WHE then 1. there is an unreduced theory (H n ; @ n ) given by

(X; A) 

H n : [f ] (Y; B ) 

H~ n (X [ CA) [f ] /

H~ n (Y [ CB ) 

@ n : H~ n (X [ CA) ! H~ n (CA=A)  = H~ n 1 (A) ; 2. (H n ; @n ) is naturally equivalent to (Hn ; @n ); 3. (H~ n ; n ) is naturally equivalent to (H~ n ; n ).

4.3. DEDUCTIONS FROM THE AXIOMS

117

Here natural equivalence means a natural equivalence of homology functors, commuting with their respective natural transformations. Proof: The details are rather lengthy, but given fully in Switzer [106], pp. 112{117.

Corollary 4.3.19 (Reduced and unreduced theories equivalent on CW )



WHE is automatically satis ed by CW-complexes, so unreduced and reduced theories are equivalent for those spaces. 

Theorem 4.3.20 (Relation of reduced and unreduced theories) If H and H~ are related both by reduction and unreduction of each other, and if  ,! X is a co bration, then Hn (X; )  = H~ n (X ) whence Hn (X )  = H~ n (X )  Hn () : Proof: The result follows from showing that (X; ) ' (X [ C ; 1); complete details

appear in Gray [38, p. 185].

Corollary 4.3.21 Hn() = H~ n(S0) = H n().

 

Next we have an important property of the ordinary homology of CW-complexes. Corollary 4.3.22 (Non-negative ordinary theories on CW ) If H is an ordinary theory and X is a connected CW-complex, then H n (X; ) = 0 for all n > 0.

Proof: Inducting from the suspension isomorphism, we obtain Hn (X; )  = Hn+m (S m X; ) : Thus for n > 0, we have H n (X; )  = H0 (S n X; )  = H~ 0 (S n X ) = 0. Here we used Corollary 4.3.15.  De nition 4.3.23 Let H be an ordinary homology theory. When each Hm(X ) is a nitely generated R-module, there is a well-de ned R-rank bm of Hm (X ) called the mth Betti number of X . When R is a eld, bm is just the dimension of the vector space Hm (X ) over R. When the homology modules are all nitely generated and there are only nitely many which are nonzero, we say that the homology of X is nite. In this case, we also de ne the Euler characteristic (number) of X as +1 X (X ) = ( 1)mbm : 1

This is traditionally applied only when the homology is non-negative, as it is for

CW-complexes. The main examples of spaces with nite ordinary homology are

the compact spaces. Each of these numbers contains progressively less information about X . However, it is enough for the solution of several important problems. This is largely due to the invariance of these numbers under homotopy equivalence (cf. on page 129).

118

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Ex 1. Use the Euler characteristic to show that a sphere cannot be homotopy equivalent to a point. There are no trivial spheres|all spheres are topological individuals; in a homological sense, every sphere is a minimally nontrivial example of its dimension class. 2. Re ect on the fact that our results for spheres are valid for spaces that are spheres only up to homology equivalence|even less than homotopy equivalence. Find some examples that distinguish between these equivalence classes.

4.3.2 Deductions from homology Actually, a number of other powerful results can now be deduced; we present some as theorems and others as guided exercises. Observe how much we can squeeze out of the identity and antipodal maps on spheres and remember that, since everything is based on homotopy type invariance, we get much more than results about spheres. In the following results, H denotes any ordinary homology theory. If we need to specify the coecients as, for example, Z we write H ( ; Z).

Theorem 4.3.24 (Brouwer xed point) Every continuous map from the closed n-ball to itself, for n  1, has a xed point. Proof: Just replace 1 by Hn in the proof for n = 1.



Theorem 4.3.25 (Degree) There is a well-de ned map deg : n (Sn) = [Sn; Sn] ! Z which preserves compositions.

Proof: De ne deg f for continuous f : Sn ! Sn to be the integer multiplier induced

as the homomorphism

f : Hn (Sn; Z)  = Z: = Z ! Hn (Sn; Z)  It is evidently a homotopy type invariant and sends composites to products of their degrees. 

Theorem 4.3.26 (Suspension preserves degree) If f : Sn ! Sn is continu-

ous, then deg f = deg(Sf ).

Proof: This is a typical Mayer-Vietoris argument. Let U denote the complement of the north pole and V the complement of the south pole in Sn+1. Note that the inclusion i : Sn ,! U \ V is a homotopy equivalence, and that the connecting map

4.3. DEDUCTIONS FROM THE AXIOMS

119


: Hn+1 (Sn+1) ! Hn (U \ V ) is an isomorphism for n  1. The following diagram commutes. Hn+1 (Sn+1)
= Hn (U \ V ) i= Hn (Sn) o

/

(Sf )

f

Hn+1 (Sn+1)
= Hn (U \ V ) i= Hn (Sn) Thus (Sf ) =
1 i f i 1
so, applying this to H ( ; Z), we get deg f = deg(Sf ) for n  1. We leave the case n = 0, when
is only monic, as an exercise.  



/



o

Corollary 4.3.27 Regarding Sn ,! Rn+1 , if f : Sn ! Sn : (x1 ; : : : ; xn+1 ) 7!

(x1 ; : : : ; xi ; : : : ; xn+1 ) then deg f = 1.

Proof: First, we reduce to the case i = 1. Let g(x1 ; : : : ; xn+1 ) = ( x1 ; : : : ; xn+1 ) and let h be the homeomorphism of Sn which interchanges x1 and xi . Then deg h = 1 and f = hgh, so deg f = (deg h)2 deg g = deg g.

Thus it suces to show deg g = 1. But this is trivial when n = 0, and g for all other n is an iterated suspension of this one. 

Theorem 4.3.28 (Antipodal degree) The degree of the antipodal map a : Sn ! Sn : x 7 ! x

is ( 1)n+1 .

Proof: Clearly a is a composition of n + 1 maps, each of degree 1.  Corollary 4.3.29 If a continuous map f : Sn ! Sn has no xed points, then deg f = ( 1)n+1 :

Proof: Such an f : Sn ! Sn is homotopic to the antipodal map by: (1 t)f (x) tx : F : Sn  I ! Sn : (x; t) 7 ! k(1 t)f (x) txk



Corollary 4.3.30 If f : S2k ! S2k and f is homotopic to the identity, then f has

a xed point.

Proof: Such an f has degree +1 = deg 1S2k. But by Corollary 4.3.29, if f has no xed points then it has degree 1 and we have a contradiction. Hence, f must have a xed point. 

120

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Theorem 4.3.31 (Hopf) Two continuous maps from Sn to itself for n  1 are homotopic if and only if they have the same degree. Proof: From the Degree Theorem we need only prove that maps having the same degree are homotopic; this is done by induction on n. For n = 1, a map with degree k is representable as a periodic real function on R which increases by k each time

its argument increases by 1. Any two such maps are homotopic. Assuming the result for n 1, two maps with the same degree admit representation as the suspensions of maps of Sn 1 to itself and suspension preserves the degree. Hence the maps we suspended are homotopic, but then so are their suspensions and the result follows. 

Ex on degree 1. If f : Sn ! Sn is constant, then deg f = 0. 2. Show that the identity map 1Sn is not homotopic to a constant map. 3. If f : Sn ! Sn is a homeomorphism, then deg f deg f

1 = deg f  f 1 = deg 1 n = 1 : S

4. Construct one representative map f : S1 ! S1 for each element in 1 (S1) and explain why only two of these maps are homeomorphisms. 5. Find a continuous f : S2 ! S2 which preserves antipodal points and show that its degree is odd. 6. Find the degrees of the following maps from Sn to Sn for n > 0: (i) a rotation about the polar axis; (ii) antipodal composed with a polar rotation; (iii) projection onto the southern hemisphere; (iv) re ection in the equator.

Theorem 4.3.32 (Borsuk's antipodal) If a continuous map f : Sn+1 ! Sn+1

preserves antipodes, then it has odd degree.

Proof: For de niteness, we note that preserving antipodes means commuting with the antipodal map. We induct on n, and the induction step is another MayerVietoris argument almost identical to the one used in the proof that suspension preserves degree, Theorem 4.3.26. Indeed, we use the same set-up and diagram, merely replacing (Sf ) there with f here and f there with f jSn here, regarding Sn ,! Sn+1 as the equator. Then we apply the conclusion from the diagram to H ( ; Z2) now instead of H ( ; Z). This reduces us to the n = 1 case, and we leave that as an exercise, along with the trivial n = 0 case.  Theorem 4.3.33 (Antipodal subsphere) If a continuous f : Sm ! Sn preserves antipodes, then m  n.

4.3. DEDUCTIONS FROM THE AXIOMS

121

Proof: Suppose that m > n > 0, and that the given f preserves antipodal pairs of points. Observe that

Sn  = S = f(x1 ; x2 ; : : : ; xm+1 ) 2 Sm  Rm+1 j xi = 0; i > ng : Put g = f jS , and observe that g : Sn ! Sn preserves antipodes so its degree is odd.

But we can homotope g to a constant map because g admits an extension over the closed upper hemisphere of Sm. So deg g = 0, a contradiction. Hence, m  n. 

Theorem 4.3.34 (Borsuk-Ulam) Every continuous f : Sn ! Rn must identify

a pair of antipodal points.

Proof: Suppose that the given f does not identify any pair of antipodal points. Then we can construct:

g : Sn ! Sn 1 : x 7 ! kff ((xx)) ff (( xx))k :

This map preserves antipodes, which contradicts the previous theorem.



Corollary 4.3.35 (Meteorology theorem) Somewhere on the Earth, there is a pair of antipodal points having simultaneously the same temperature and pressure. Proof: Let T; P respectively denote the temperature and pressure functions, assumed to be continuous, on the Earth's surface. Then we have a continuous map f : S2 ! R2 : x 7 ! (T (x); P (x)) and the theorem applies.



The next result is a corollary that explains the problems of gift-wrapping a soccer ball or making maps of the world.

Theorem 4.3.36 (Gift wrap) No subspace of Rn can be homeomorphic to Sn. Proof: By the Borsuk-Ulam Theorem, no map from Sn to Rn can be injective if it is continuous.  Theorem 4.3.37 (First hairy ball) An n-sphere admits a continuous nowherezero tangent vector eld if and only if n is odd.

Proof: For n = 2k even, suppose that we have a continuous nowhere-zero vector v : S2k ! R2k+1 n f0g : We shall show that it cannot be everywhere tangent to S2k. Construct eld

f : S2k ! S2k : x 7 ! kvv((xx))k :

122

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Now, this f is continuous because v is, and moreover, f is homotopic to the identity 1S2k. Hence, by Corollary 4.3.30, f has a xed point at some x0 . Then v(x0 ) = x0 , and so v is not tangent but normal at x0 . For odd n = 2k 1, we construct v : S2k 1 ! R2k : (x1 ; : : : ; x2k ) 7 ! ( xk+1 ; : : : ; x2k ; x1 ; : : : ; xk ) : This is evidently continuous and nowhere zero; also, it is everywhere orthogonal to the normal and hence a tangent eld. 

Theorem 4.3.38 (Ham sandwich) Given bounded subsets Ai for i = 1; : : : ; n in Rn , then there exists a hyperplane which simultaneously bisects all of the Ai .

The name comes from the case n = 3: two pieces of bread and a slice of ham, stacked in any way, can be fairly divided with one cut. Proof: Take any x 2 Sn 1  Rn and construct the tangent space there, TxSn 1. In Rn , nd that hyperplane Px which is parallel to TxSn 1 and which also bisects A1 . De ne Ai (x) to be the measure of Ai on the outside of Px . Observe that the outside of Px is the inside of P x . Next construct the continuous map f : Sn 1 ! Rn 1 : x 7 ! (A2 (x); A3 (x); : : : ; An (x)) : By the Borsuk-Ulam Theorem, there exists x0 2 Sn 1 with f (x0 ) = f ( x0 ), so we choose Px0 .  Now consider wrapping a soccer ball with three pieces of paper; we nd that one piece must cover a pair of antipodal points. Theorem 4.3.39 (Lusternik-Schnirelmann) If Sn is covered by n + 1 closed sets A1 ; : : : ; An+1 , then one of them must contain an antipodal pair.

Proof: The union of the given sets is Sn  Rn+1 , so we construct the continuous map f : Sn ! Rn : x 7 ! (d(x; A1 ); : : : ; d(x; An )) :

This must identify a pair of antipodal points, x0 and x0 say, by the Borsuk-Ulam Theorem. Now there are only two possibilities: either d(x0 ; Ai ) = 0 for some i, whereupon x0 ; x0 2 Ai because Ai is closed; or d(x0 ; Ai ) > 0 for all i = 1; : : : ; n, whence x0 ; x0 2 An+1 because the Ai form a cover. 

4.3.3 The Lefschetz theorem We have seen how useful the Euler characteristic is in comparing spaces and just saw how the degree of a map from a sphere to itself could solve otherwise apparently dicult problems. Next we come to another characteristic, the Lefschetz number, for a continuous map from a space with nite homology to itself. A sphere is of course a special case, and we shall see that for spheres the new number relates to the degree. We obtain also a powerful xed-point theorem with useful applications.

4.3. DEDUCTIONS FROM THE AXIOMS

123

Theorem 4.3.40 (Lefschetz number) Let X be a space with nite homology. There is a well-de ned map, from the set [X; X ] of homotopy classes of continuous self-maps on X , to the rationals Q :  : [X; X ] ! Q : [f ] 7! f :

Proof: For all k 2 Z, there is induced a linear map on rational homology fk : Hk (X ; Q ) ! Hk (X ; Q ) : We recall from linear algebra that the trace is independent of the choice of basis. We de ne 1 X f = ( 1)k tr fk : Clearly f is well de ned.

k= 1



Our next theorem shows that f is the `obstruction' to f being xed-point free.

Theorem 4.3.41 (Lefschetz xed point) Let f : X ! X be a continuous map from a compact space to itself and let f be the Lefschetz number of f . Then it follows that (i) if f 6= 0, then f has a xed point; (ii) if X has the rational homology of a point, then X has the xed point property; (iii) if 1X is homotopic to f and f has no xed point, then the Euler characteristic of X is zero; (iv) if X = Sn, then

f = 1 + ( 1)n deg f ;

(v) if the Euler characteristic of X is nonzero, then every ow

 : X  R ! X : (x; t) 7 ! (x; t) = t (x) ; on X , has a xed point x0 2 X with t (x0 ) = x0 for all t 2 R.

Proof: (i) A proof for simplicial complexes is given on p. 163. General proofs may

be found in Gray [38, Sect. 26] or Vick [114, Chapt. 6]. (ii) By hypothesis, X has only one component so the only nonzero rational homology is H0 (X ; Q ) = Q . It follows that f0 = 1Q, whence  = ( 1)0 tr f0 = 1 6= 0 and we have an obstruction to f being xed-point free, as required.

124

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

(iii) Suppose that f is xed-point free, so f = 0, and homotopic to 1X . It induces identity morphisms on homology and the trace of an identity linear map is the dimension of its domain: f =

n X k=0

( 1)k tr(1k ) =

n X k=0

( 1)k dim Hk (X ; Q ) = (X ) :

(iv) In the case of a continuous f : Sn ! Sn, we have

H0 (Sn; Q )  = Hn (Sn; Q )  =Q

f = tr(f0 ) + ( 1)n tr(fn ) = 1 + ( 1)n deg f : (v) A ow is by de nition continuous, each t is a self homeomorphism of X and satis es the conditions 0 = 1X and t1 +t2 = t1  t2 : First we see that every t is homotopic to 1X by

F : X  I : (x; s) 7 ! ((1 s)t; x) : So f = 1X = (X ) 6= 0 and hence each t has a xed point. We need to show that there is a xed point common to all; we use induction and the nite intersection property as follows. For each natural n, de ne Xn to be the (nonempty, closed) set of xed points of 1=2n . Now, X1 = \n Xn 6= ?, because Xn+1  Xn . So, X1 is a set of points xed under all rationals of dyadic form k=2n; but, as we exploited in the proof of the simplicial approximation theorem, such rationals are dense in the set of reals. It follows that every element in X1 is xed under t for all t 2 R. 

Corollary 4.3.42 We note the following easy deductions: 1. 1X = (X ). 2. The antipodal map a : Sn ! Sn : x 7! x has

a = 1 + ( 1)n ( 1)n+1 = 1 + ( 1)2n+1 = 0 : 3. If f : Sn ! Sn is not a homeomorphism, then its degree is not 1; hence f 6= 0 and it must have a xed point. 

Ex 1. Give another proof of the Brouwer Fixed Point Theorem using the Lefschetz Fixed Point Theorem.

4.4. HOMOLOGY OF CHAIN COMPLEXES

125

2. Show that RP 2 has the xed point property, as in fact do all RP 2m because

Hk (RP 2m ; Z)  = 0 or Z2 if k > 0 :

3. 4. 5. 6. 7.

8.

However, RP 1  = S1 certainly does not. Regarding S2n  R2n+1 , no continuous f : S2n ! S2n has f (x) orthogonal to x for all x 2 S2n. Find the degree of all xed-point free maps from Sn to itself and deduce that if n is even, then no such map can be homotopic to the identity. Up to isomorphism there are only two groups with free actions on evendimensional spheres, because maps on them homotopic to the identity are not xed-point free. If f : S2n ! S2n is continuous, then either f has a xed point, or f  f has a xed point and f sends some point to its antipode. The only compact closed surfaces with Euler characteristic zero are the Klein bottle, which is nonorientable, and the torus, which is orientable. Only these two admit a xed-point free map that is homotopic to the identity. On the torus, a xed-point free map can be obtained by a ow along a nowhere-zero tangent vector eld. Open-ended problem: Construct a sequence of maps f0; f1; : : : with increasing (decreasing) Lefschetz number, starting with f0 = 0, and investigate the corresponding sequence of numbers of xed points. [Hint: A convenient domain to start with would be S2; for a decreasing sequence of Lefschetz numbers try S1.]

4.4 Homology of chain complexes As will be anticipated, the axioms for homology can be realized in a variety of ways depending on the particular topological subcategory that is of interest. Thus, particular features or extra structure on spaces can be exploited in dierent ways to yield equivalent homology theories. For example, simple compact spaces, especially surfaces, can be dealt with quite eciently by utilizing families of free abelian groups generated by simplices of dierent dimensions. This approach to triangulable spaces yields simplicial homology and a good text suitable for a rst course in it is provided by Armstrong [2]. At the other extreme, we may have potentially sophisticated spaces with considerable extra structure, such as bundles over dierential manifolds. Then it is convenient to formulate a cohomology theory in terms of free abelian groups generated by dierential forms. This yields de Rham cohomology and an introductory treatment can be found in Singer and Thorpe [95] or Warner [116]. See Appendix E for some computational help.

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

126

In both of these examples, and in many others, the construction follows a standard pattern: use whatever features are convenient to construct a sequence of abelian groups and homomorphisms that form a chain complex. At each group there is a unique quotient group determined and it turns out to have the desired properties for a homology (or a cohomology) theory provided that we have properly chosen our chain complex. Thus a construction problem is reduced to the synthesis of appropriate chain complexes. We begin with a study of these entities and follow with examples. Later we shall construct homology and cohomology theories by much more general methods. Denote by C a xed subcategory of the category Ab of abelian groups and group homomorphisms.

De nition 4.4.1 A C -chain complex is a set K = fKm; @m j m 2 Zg m K of objects and morphisms Km @! m 1 from C satisfying @m @m+1 = 0. Each @m is called a boundary operator.

Hence, K can be usefully viewed as a doubly in nite sequence diagram in C






K:

o

@ 1

K

1

@0 o

K0

@1 o

K1 o

@2






with the morphism conditions im @m+1  ker @m for all m 2 Z. As we have remarked, the case where each K is simply an abelian group is basic to our subject. However, in the sequel we shall study also the case when, for some xed commutative ring R with identity, each Km is a module over R. In particular the rings Z, Zk, Q , R, and C should be borne in mind. Of course, the category of Z-modules can be identi ed with Ab itself.

De nition 4.4.2 A C -chain map f between two C -chain complexes K and K 0 is a set ffm : Km ! Km0 g of C -morphisms that satisfy, for all m 2 Z, @m0 fm = fm 1 @m :

Then by f : K ! K 0 will be understood the commuting diagram






@ 1 o

K

1

@0 o

f 1




o

@0 1

K0 o

@1

f0

K0 1 

o

@00

K1 o

@2






@20






f1

K00 

o

@10

K10 

o

4.4. HOMOLOGY OF CHAIN COMPLEXES

127

Theorem 4.4.3 (Category of C -chain complexes) The C -chain complexes and their C -chain maps with natural composition form a category C hain.  De nition 4.4.4 The homology H (K ) of a chain complex K is the sequence of



quotient objects





Hm (K ) = imker@ @m m 2 Z : m+1

It is fundamental that homology measures the departure of the chain complex from exactness. Betti numbers and Euler characteristics of chain complexes are de ned as for spaces. Each of these numbers contains progressively less information about K ; however, it is enough for the solution of several important problems. This is largely due to the invariance of these numbers under an appropriate equivalence relation on chain complexes, as we shall see shortly (cf. on page 129). Theorem 4.4.5 (C -chain homology) Every C -chain map f : K ! K 0 determines a homology map H (f ) consisting of a set of C -morphisms Hm (f ) : Hm (K ) ! Hm (K 0 ) : c + @m+1 Km+1 7 ! fm c + @m0 +1 Km0 +1 where c 2 ker @m . It follows that for each m we have a covariant functor Hm (K ) K

Hm : C hain /

f 

C:



Hm (f ) /

Hm (K 0 )

K0





Subsequently we shall be interested in calculating the homology of chain complexes associated with topological spaces. Various choices are usually available for chain complexes, some geometrical and others more analytical. However, given some choice, we need to know which chain maps give rise to the same homology maps; this follows from the algebraic notion of chain homotopy. We shall see later that the reason for this name is the induction of such algebraic maps by topological homotopies. De nition 4.4.6 A C -chain homotopy s between two c-chain maps f; g : K ! K 0 is a set fsm : Km ! Km0 +1 jm 2 Zg of C -morphisms




BB o






@ 1

K

1B o

@0

K0 A @1 K1 @ @2






K00






o

o

AA @ B BB BB f 1 g 1 BBsB 1 f0 g0 AsA0 f1 g1 @@@s1 AA BB s 2 BB @@ B A B @

K0 1 !

o

@0 1







!

o

@00



K10 

o

@10



o

@20

128

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

satisfying for all m 2 Z

@m0 +1 sm + sm 1 @m = fm gm ; and if such a set exists we write s : f  g. A C -chain complex K is called chain

contractible if there is a chain homotopy from the identity 1K to the zero chain map on K .

Theorem 4.4.7 (Chain contractibility) A chain complex K composed of free abelian groups (a free chain complex) is chain contractible if and only if it has trivial homology (then we say it is acyclic). Proof: Suppose we have a chain homotopy s : 1  0; then (8 m 2 Z) @m+1 sm + sm 1 @m = 1 0 = 1 ; Hm (K ) = imker@ @m : m+1 Now, if @m (x) = 0, then @m+1 sm (x) = x, so x 2 im @m+1 and ker @m  im @m+1 . But @m @m+1 = 0 so im @m+1  ker @m and for all m 2 Z, Hm (K ) = 0 : Conversely, if for all m 2 Z then we de ne

ker @m = im @m+1 ;



0; sm : Km ! Km+1 : x 7 ! 0y ifif @@m xx 6= x m = 0;

where @m+1 yx = x. Hence, for any x 2 Km , either @m x = 0, when (@m+1 sm + sm 1 @m )(x) = x ;

or @mx = z 6= 0, when (@m+1 sm + sm 1 @x )(x) = sm 1 (z ) = x : If also @m x0 = @m x 6= 0, then (x x0 ) 2 ker @m ; but, since neither x nor x0 is in ker @m , it follows that x = x0 . Hence s is well de ned and also a chain homotopy.



Theorem 4.4.8 (Chain homotopy invariance) Chain homotopic maps induce

the same maps in homology:

s : f  g =) H (f ) = H (g) :



4.4. HOMOLOGY OF CHAIN COMPLEXES

129

Ex on chain homotopy Observe that s gives commuting diagonal maps in the diagram of commuting squares generated by f and g.

The functorial property of H on morphisms preserves isomorphisms, but we see next that in order to have isomorphisms in homology it is sucient to have chain complexes equivalent up to chain homotopy.

Theorem 4.4.9 (Chain homotopy equivalence) If f : K ! K 0 and h : K 0 ! K satisfy fh  1K 0 and hf  1K , then H (f ) and H (h) consist of isomorphisms in C .  Corollary 4.4.10 Chain homotopy equivalent chain complexes have the same Betti numbers, and hence the same Euler characteristic (when de ned). (Cf. page 84, page 117, and page 162.)  Just as the theory of vector spaces is enriched by the study of dual spaces, so also is homology theory augmented by its dual, cohomology. The dual to a vector space is the usual vector space of scalar-valued linear maps on the rst space. For a chain complex K of R-modules, the natural choice of dual uses R-valued module homomorphisms on the Km; that is, HomR (Km ; R). However, we can obtain useful

exibility by working more generally with HomR (Km ; G) where G is an arbitrary Rmodule. We note that HomR (Km ; G) is an abelian group and indeed an R-module if R is commutative. More generally still, we can work with the abelian groups Hom(Km ; G) when Km and G are objects from C , our xed subcategory of Ab.

Ex on Hom 1. HomR (A; ) is a functor and HomR ( ; B ) is a cofunctor.

Q

Q

Q

2. HomR (Ai ; B )  = Hom(Ai ; B ) and HomR (A; Bi )  = Hom(A; Bi ). 3. HomZ(Z; Z2)  = Z2.

4. There is a short exact sequence of abelian groups

Z2 0 Z  Z where  : k 7! 2k and : 2k 7! 0. One may deduce that HomR ( ; R) does not 0

/

/

/

/

preserve exact sequences by investigating exactness in 0 o

HomZ(Z; Z2) o

 Hom (Z; Z ) 2 Z

 o

HomZ(Z2; Z2)

5. For any abelian group G, show that there is an isomorphism Hom(Z; G) ! G :  7 ! (1) :

o

0:

130

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

6. For any m; n 2 N , show that there is a homomorphism Hom(Zm; Zn) ! Zn :  7 ! (1) which induces an isomorphism Hom(Zm; Zn)  = Z(m;n) where (m; n) denotes the greatest common division of m and n.

De nition 4.4.11 Given a C -chain complex K = fKm; @m j m 2 Zg and a xed object G in C we de ne the C -cochain complex K  of K with coecients in G to be (Hom(K; G); ) = fK m; m j m 2 Zg where for each m, K m = Hom(Km ; G) and m = Hom(@m+1 ; 1G ) : K m ! K m+1 : f 7 ! f  @m+1 :

Km;

o

@m+1

;; ; f ;; ; 

Km+1 m (f )

G



Theorem 4.4.12 (Coboundary operators) The m are C -morphisms satisfying

the condition

m+1 m = 0 :

The m are called the coboundary operators.  De nition 4.4.13 The cohomology H (K ; G) of the C -cochain complex K with coecients in G is the set of C -objects   m H m (K ; G) = imkerm 1 m 2 Z : The notation H m (K ; G) = H m (Hom(K; G)) is also used. When G = Z it is frequently suppressed: H m(K ) = H m (K ; Z). A C -chain complex is equivalently de ned as a graded C -object fKmjm 2 Zg with a dierential @ of degree 1 called the boundary operator. Then a C cochain complex is a graded C -object with a dierential  of degree +1 called the coboundary operator. Every such cochain complex determines a unique chain complex by changing m to m in its indexing set. By this device, we obtain the notions of cochain map, cochain category, and cochain homotopy with properties dual to those arising from chains. Then for each m, H m ( ; G) is a cofunctor. The cohomology of a cochain complex is the homology of its associated chain complex. However, it turns out that cohomology is in fact richer in structure than homology by possessing a natural product, so we shall have cohomology rings. This, for example, will allow us to distinguish between S4  S2 and C P 3 although they have the same homology and cohomology groups.

4.4. HOMOLOGY OF CHAIN COMPLEXES

131

Ex on Betti numbers for cohomology Betti numbers and the Euler characteristic are de ned similarly for cohomology. The Euler characteristic is the same in both cases.

The following terminology is standard for any C -chain complex K = fKm; @m j m 2 Zg ; elements of Zm (K ) = ker @m  Km are called m-cycles; elements of Bm (K ) = im @m+1  Km are called m-boundaries; two m-cycles in the same homology class are called homologous if and only if they dier by an m-boundary. Correspondingly, for any C -cochain complex, we have its m-cocycles and mcoboundaries. However, in de Rham cohomology an older terminology has persisted: m-cocycles are called closed m-forms and m-coboundaries are called exact m-forms ; we return to this later (page 167 et seq.). A technique for constructing homology and cohomology theories is now clear enough. We need an appropriate C -chain complex to utilize its homology, so ensuring functoriality. The appropriate topological homotopies must be shown to induce chain homotopies. We can often establish the required exactness property from the following result. Theorem 4.4.14 (Short exact sequence functor) There is a covariant functor from the category of short exact sequences of C -chain complexes to the category of (long) exact sequences of groups. It sends each diagram 0 /

A f



/

0

/

C

0 /



A0 f 0 B 0 g0 C 0 

/

g

B 



/

/

/

0

commuting in C hain with exact rows, to the commutative homology diagram with exact rows




/

Hn (A) f Hn (B ) g Hn (C ) @ Hn 1 (A) /



/





/






/






Hn (A0 ) f0 Hn (B 0 ) g0 Hn (C 0 ) @0 Hn 1 (A0 ) Proof: The crucial map @ is called the connecting morphism and is de ned by @ : Hn (C ) ! Hn 1 (A) : [z ] 7 ! [xz ] ; where the exactness in the chain rows is exploited by putting z = g(d) for some d 2 B (since g is surjective) and @ d = fxz for the unique xz 2 A (since f is injective). Of course, f in the diagram simply abbreviates Hn (f ), etc. 




/



/



/





/

/

132

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Ex on chain complexes 1. Any abelian group G determines a trivial chain complex with Km = G and @m = 0 for all m 2 Z. Hence, given any nite diagram

K0 @1 K1 @2 K2 @3


@n Kn o

o

o

o

in Ab with @m @m+1 = 0, we can extend it to be a chain complex with Km = 0 and @m = 0 outside the given diagram. 2. Ab is isomorphic to the category of Z-modules. 3. C -chain complexes and C -chain maps form a category. 4. Given a sequence fGm j m 2 Zg of abelian groups, does there always exist a chain complex K with homology Hm (K ) = Gm for all m 2 Z? 5. Write out enough details to de ne C -cochain maps, the category of C -cochains and C -cochain homotopy, listing explicitly the properties dual to those found for chain complexes. 6. A chain complex is an exact sequence if and only if it has trivial homology. 7. Formulate a de nition for exactness of a sequence of C -chain complexes and show that short exact sequences of C -chain complexes form a category. 8. Verify the exactness of the homology sequence arising from a short exact sequence of chain complexes. We saw above that cohomology H m (K ; G) is de ned with the cochain complex m fK ; m g, where K m = Hom(Km ; G) and m = Hom(@m+1 ; 1G ), constructed from the chain complex fKm; @m g. The use of Hom means that there is a natural pairing h ; i : H m (K ; G) Hm (K ) ! G with respect to which  and @ are adjoints:

hu; i = hu; @i : This pairing is called the Kronecker product (formerly, index). In terms of a representative cocycle u 2 Hom(Km; G) and a representative cycle v 2 Km, h[u]; [v]i = u(v) 2 G :

Ex on Kronecker product h ; i is a well-de ned pairing on H m(K ; G) Hm(K ):

that is, the element obtained in G does not depend on the choice of representatives.

4.4. HOMOLOGY OF CHAIN COMPLEXES

133

4.4.1 Universal coecient theorems The Kronecker product appears in one of several Universal Coecient Theorems, which name indicates that knowledge of ordinary (co)homology with integer coef cients suces to determine ordinary (co)homology with any coecient group G. In the literature, it is most common simply to cite `a UCT' and to leave the reader to decide precisely which one is being used. We now consider a few of the simplest, one of which is generally adequate.

Theorem 4.4.15 (Universal coecient theorem 1) There is a natural short

exact sequence

;i 0 ! Ext(Hm 1 (K ); G) ! H m (K ; G) h! Hom(Hm (K ); G) ! 0 ; where h ; i denotes the Kronecker product, which splits but unnaturally.

Proof: Ext( ; ) is the extension functor de ned as follows. Let A and B be abelian groups and consider a free resolution of A 0 ! R ! F ! A ! 0: That is, a short exact sequence in which F is free and R stands for the relations, so that A  = F=R. Then we de ne Ext(A; B ) = coker (Hom(F; B ) ! Hom(R; B ))  = Hom(R; B )= im Hom(F; B ) ;

so that Ext( ; B ) measures the failure of Hom( ; B ) to be exact. See Gray [38, Sect. 25] for more details or Vick [114] for hints.  Here are some important properties of Ext.

Ex on the extension functor Ext 1. Ext(A; B ) = 0 if A is free or if B is divisible. We recall that B is divisible if and only if uB = B for every integer u  1. 2. The eld of rationals is divisible as an abelian group, as is n Zp1 = nlim !1 Zp

where p is any prime. B is divisible if it is a direct sum of various Zp1 and copies of the rationals Q . It follows that R and C are divisible. 3. Ext(Zp; Zq)  = Z(p;q) where ( ; ) denotes the least common multiple. More generally, Ext(Zp; B )  = B=pB . 4. Ext(i Ai ; B )  = i Ext(Ai ; B ). It may be helpful to recall that for nite collections, direct sum  and direct product  coincide.

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CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

5. Ext(A; i Bi )  = i Ext(A; Bi ). 6. Ext(A; B ) is the group of all equivalence classes of extensions of A by B ; that is, of all classes of short exact sequences 0 !B !C !A !0 with the group operation derived from direct sum and the equivalence relation from the 5-lemma. 7. Always, H 1 (K )  = H1 (K ) and H 1 (K ; G)  = Hom(H1 (K ); G).

Theorem 4.4.16 (Universal coecient theorem 2) If each H m(K ) is nitely

generated or if G is nitely generated, then there is a natural short exact sequence 0 ! H m(K ) G ! H m (K ; G) ! Tor(H m+1 (K ); G) ! 0 ; which splits, but unnaturally. If instead each Hm (K ) is nitely generated or if G is nitely generated, then there is a natural short exact sequence 0 ! Tor(Hm 1 (K ); G) ! Hm (K ; G) ! Hm (K ) G ! 0 ; which splits, but unnaturally.

Proof: Here Tor( ; ) is the torsion functor, and Tor( ; B) similarly measures the failure of B to be exact. Thus, if 0 !R !F !A !0 is a free resolution as before, then Tor(A; B ) = ker(R B ! F B ) : For further details see Gray [38] or Vick [114].  We present the companion list of properties of Tor.

Ex on the torsion functor Tor 1. Tor(A; B )  = Tor(B; A). 2. 3. 4. 5.

Tor(A; B ) = 0 if either A or B is free. Tor(Zp; Zq)  = Z(p;q) where (p; q) denotes the greatest common divisor of p; q. More generally, Tor(Zp; B )  = fb 2 B jpb = 0g. Tor commutes with direct sums: Tor(; )  =  Tor( ; ) Tor( ; )  =  Tor( ; ):

4.5. HOMOLOGY AND COHOMOLOGY OF CW-COMPLEXES

135

In general, Tor is much easier to calculate than Ext; of course, Ext is more important in abstract homological algebra. One other place where Tor is needed, however, is the Kunneth formula for the (co)homology of a product.

Theorem 4.4.17 (Kunneth formula) If K and K 0 are free C -chain complexes,

then

Hn (K K 0 )  =

X p+q=n

Hp (K ) Hq (K 0 ) 

X r+s=n 1

Tor (Hr (K ); Hs (K 0 )) :

See Vick [114, pp. 98{102] for a proof.

Ex Write down the version for cohomology: just raise the indices.

4.5 Homology and cohomology of CW-complexes As we have anticipated earlier, the CW-decomposition of a space does indeed allow a simple calculation of ordinary homology and cohomology groups (cf. Lundell and Weingram [64]). The basic free abelian groups are generated by the m-cells, as we might have hoped; hence they are found with much less work than will be the case for simplicial complexes. The family of boundary homomorphisms depends on the isomorphism m (Sm; s0 )  = Z, to incorporate the degree of a map into a formal sum of generators. Now we use the homotopy group n (Sn) of a sphere Sn to rede ne degree in a way that makes it directly available to index the boundaries of cells via their attaching maps. The role of the integer degree for cells corresponds to some extent with the use of an integer from Z2 to indicate the orientation of a simplex. We summarize the construction, then compute the homology of some familiar spaces. Given a CW-complex structure

fe ;  j  2 J g

on a Top space X , denote by Cm (X ) the free abelian group generated by

Jm = f 2 J j e is an m-cellg for m 2 Z. Observe Cm (X ) = 0 for m < 0. For each m 2 Z we de ne, by linearity over generators,

@m : Cm (X ) ! Cm 1 (X ) :  7 ! Here,

X 2J m 1

deg(h ) :

h : (Sm 1; s0 ) ! (Sm 1; s0 )

is obtained from the restriction of the attaching map

 : (B m ; Sm 1; s0 ) ! (X m ; X m 1; x0 )

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CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

to the boundary e_ , which we shall use via the homeomorphism e_   = Sm 1:

Sm

1

h /

Sm

1

=



Xm 

1

B m 1= Sm 

/

/

2

where, with e denoting the interior of e ,

Xm

1

m 1 ! ! X mX 1 n e = B m 1= Sm

2

and we have omitted basepoints. In the diagram, X m 1 denotes the (m 1)-skeleton of X , the union of cells with dimension not exceeding (m 1), and X m = ? if m < 0; nally, for all m > 1, deg(h ) = [h ] 2 Z  = m 1 (Sm 1) :

Ex Repeat the Ex on degree (page 120) as much as possible. In particular, nd the degree of your favorite maps on spheres, including

deg(homeo) = 1 ; deg(const) = 0 : We recall that for any CW-complex, the adjacent skeletons are related by

X m =X m 1  =

_

2Jm

Sm :

De nition 4.5.1 A homology theory H is said to satisfy the wedge axiom if inclusions

8 9 < _ = :Y ,! 2M Y;

in Top

induce isomorphisms on H for all m 2 Z:

M 2M

0 1 _ Hm (Y )  = Hm @ Y  A : 2M

Evidently, if we wish to have coecients in a group G then we can replace Cm (X ) by Cm (X ) G and @m by @m 1. The properties are summarized in the following.

Theorem 4.5.2 (Natural wedge equivalence) We have: 1. fCm(X ) G; @m 1 j m 2 Zg is a chain complex C (X ) G.

4.5. HOMOLOGY AND COHOMOLOGY OF CW-COMPLEXES

137

2. The homology of C (X ) G is naturally equivalent to any ordinary homology theory H with coecients G that satis es the wedge axiom. 3. The natural equivalence of homology arises from a natural equivalence

Hm (X m =X m 1) ! Cm (X ) G ; Hk (X m=X m 1) = 0 if k 6= m : 4. Hm (X m) ! Hm (X ) is an epimorphism and Hk (X m ) ! Hk (X ) is an isomorphism for k < m. 5. Every cellular map f : X ! Y in CW  induces a map fm : Cm (X ) ! Cm (Y ) :  7 !

X

2Jm0

deg(f  ) ;

where f  : Sm ! Sm is the composite of quotient maps induced by f :

Sm

f  /

Sm O

=

=

X m=(X m n e ) 

/

Y m =(Y m n e )

and hence a corresponding homomorphism in homology.

Proof: Switzer [106], pp. 174{180, and Lundell and Weingram [64], pp. 148{151.  In the construction of @m above, the degree isomorphism actually provides an indexing of how the boundary, e_  = Sm 1, of each m-cell e is `stretched' across each (m l)-cell in the (m l)-skeleton. Similarly, we obtain each fm by an indexing of the behavior of a cellular map between spaces. Next we oer some practice to bring this and other features home. Just as in physics every new quantum theory development is illustrated on (and sometimes only on!) the poor old harmonic oscillator, so we rediscover again the hole in a circle as the rst exercise. Actually, this is an entirely proper state of aairs in both physics and mathematics because of the fundamental importance of the two examples. Indeed, the circle is the rst nontrivial topological space as distinct from set, and the exponential function is by far the most important function in all mathematics.

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CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES 1

e1 -e12

6

e2

-1 e1

e12

6

 X1  = S1 _ S1

X = X2

Figure 4.2: S1  S1 as a CW-complex with 1-skeleton S1 _ S1

Ex on computing homology of CW-complexes 1. The circle S1 admits a CW-complex structure fe0; e1 ; 0 ; 1 g with attaching maps: 0 : B 0 ! S1 : e0 7 ! s0 1 : (B 1 ; S0; s0 ) ! (S1; s0 ) : x 7 ! cos x + i sin x: Evidently C0 (S1)  =Z = C1 (S1), and otherwise Ck (S1) = 0. Now, ker @0 = 1 C0 (S ) and im @1 = 0. Hence H0 (S1)  = Z. Next, ker @1  = f 2 C1 (S1) j deg h = 0g = C1 (S1)  =Z and im @2  = 0; hence H1 (S1)  = Z and Hk (S1) = 0 for k > 1: 2. The torus X = S1  S1 admits a CW-complex structure

fe0; e11 ; e12 ; e2 ; 0 ; 11 ; 12 ; 2 g which corresponds with our usual plane rectangular representation with edge identi cations. The 2-skeleton X 2 is the whole space and X 1 is the one point union of two circles. ker @0 = C0 (X )  = Z and im @1 = 0 so H0 (X )  = Z:

C1 (X )  = Z  Z so ker @1 = f(; ) 2 Z  Z j  deg h1 +  deg h2 = 0g where the pointed maps h1 ; h2 : S0 ! S0 are induced by 11 and 12 , respectively. But h1 and h2 have degree zero since they are constant, so ker @1 = Z  Z :

4.5. HOMOLOGY AND COHOMOLOGY OF CW-COMPLEXES

139

im @2  = f( deg h21 ;  deg h22 ) 2 Z  Z j ;  2 Zg where h21 and h22 are the maps making the following diagram commutative: h21

e_ 2  = S1

/

S1

h22 o

e_ 2  = S1

=

2

X1  = S1 _ S1

2

S1 _ fs0 g  = fs0 g _ S1 = B 1= S0  



/

X1  = S1 _ S1 

o

Now, both h21 and h22 wrap the boundary of e2 twice round their image space S1, once forward and once backward; therefore they both have degree zero. Hence H1 (X )  = Z  Z.  Also, H2 (X ) = Z because im @3 = 0 =) H2 (X ) = ker @2  C2 (X )  =Z

and

ker @2  = f 2 Z j (deg h21 + deg h22 ) = 0g  = Z:

In fact, these computations extend easily to Sn and to Sn  Sk because of the simple way that the cells are attached together in these spaces. The wedge axiom itself deals with Sn _ Sk. 3. The real projective plane RP 2 admits a CW-complex structure

fe0; e1 ; e2 ; 0 ; 1 ; 2 g : Here the 1-skeleton is homeomorphic to S1. As before, we get H0 (X ) = Z  Z. and C1 (X ) = Now, however, 1 induces h1 : S0 ! S0 and so ker @1 = f 2 Z j  deg h1 = 0g  = Z: Also where h2 is given by

im @2 = f deg h2 j  2 Zg h e_2  = S1 2 /

S1 O

=

2

S1  = X1

X1 But h2 wraps the boundary of e2 twice round X 1  = S1 so deg h2 = 2. Hence 

/

im @2 = f2 j  2 Zg

CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

140

e0



e1

?

e2

e0

6e1 -

0 e

e1



X1  = S1

-

e1

X = X2

Figure 4.3: Projective plane as a CW-complex with 1-skeleton S1 so

H1 (X )  = Z2 : ker @2  = f 2 Z j  deg h2 = 0g = f 2 Z j 2 = 0g = 0 : Hence H2 (X )  = 0. 4. Compute H (S2) and H (Klein bottle); then compute H (Sn) for n  2. We collect these elementary results and others in Table 4.1; it is customary to denote the direct sum of k copies of Z by kZ. In each case here the space is connected (so H0  = Z), and unless indicated otherwise the top homology group is also  = Z.

Ex Verify as many as possible. [Hints: For SnSm, recall how Sn_Sm ,! SnSm; we suggest you start with S1  S2 and note that S1  = B 1= S0. For the projective spaces, you will need the attaching maps.]

In our preliminary deductions from the axioms, we showed that the following

CW-complex situations induce isomorphisms in homology: (i) inclusion of a relative CW-complex: (A; A \ B ) ,! (A [ B; B ) ; (ii) projection onto a quotient by a subcomplex: (X; A) ! (X=A; ) :

4.5. HOMOLOGY AND COHOMOLOGY OF CW-COMPLEXES

Space Rn Bn Sn Sn _ Sm Sn  Sn Sn  Sm; n 6= m RP 2 n RP 2n+1 CPn HPn

CW-cells en e0 ; en 1 ; en e0 ; en e0 ; en ; em e0 ; en1 ; en2 ; en+n e0 ; en ; em; en+m e0 ; e1 ; : : : ; e 2 n e0 ; e1 ; : : : ; e2n+1 e0 ; e2 ; e4 ; : : : ; e 2 n e0 ; e4 ; e8 ; : : : ; e 4 n Klein bottle e0 ; e11 ; e12; e2 orientable p-surface e0 ; e11 ; e12; : : : ; e12p ; e2 nonorientable p-surface e0 ; e11 ; e12; : : : ; e12p ; e2

141

Nontrivial Hk (X ; Z); k > 0

H = 0 H = 0 Hn  =Z  Hk = Hk (Sn)  Hk (Sm) Hn  = Z Z Hn  =Z = Hm  H2n = 0; H2k+1  = Z2; 0  k < n  H2k+1 = Z2; 0  k < n H2 k  = Z; 1  k  n H4 k  = Z; 1  k  n  H1 = Z  Z2; H2 = 0 H1  = 2pZ H1  = (p 1)Z  Z2, H2 = 0

Table 4.1: Homology of some simple spaces In these, (ii) is a special case of the property enjoyed by all co brations, and both extend to cohomology theories (cf. Maunder [68], p. 314, for the cohomology version of (i)). Since we obtain cohomology with coecients G as the homology of a dualized chain complex, we do not expect to gain any new information except via the product in cohomology. See sections 5.3 and 5.11 for details about the cup product. We have claimed that homology gives a useful approximation to homotopy groups. The following theorem shows that the approximation is excellent|as far as it goes!

Theorem 4.5.3 (Hurewicz isomorphism) If X is an (n 1)-connected space for some n  2 and H is ordinary homology theory, then there is a homomorphism  hm : m (X ) ! Hm (X ; Z) : an isomorphism for m  n; an epimorphism for m = n + 1:

If X is path-connected, then there is an epimorphism

h1 : 1 (X ) ! H1 (X ; Z) : We shall indicate a proof via spectral sequences in Section 6.3, beginning on page 197; see also Switzer [106], p. 185, or Maunder [68], p. 323.

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CHAPTER 4. HOMOLOGY AND COHOMOLOGY THEORIES

Corollary 4.5.4 For path-connected spaces, the homomorphism h1 induces an isomorphism 1 (X )= ker h1 = H1 (X ; Z) with ker h1 the commutator subgroup of 1 (X ), so H1 (X ; Z) is 1 (X ) abelianized. 

Chapter 5

Examples in Homology and Cohomology Proceed, proceed. We will begin these rites, As we do trust they'll end in true delights. |Shakespeare, As You Like It.

In this chapter we outline the construction of a number of frequently occurring theories. As pointed out before, going from a chain complex to homology and the dual situation of going from a cochain complex to cohomology are automatic steps; it is in the construction of the chain and cochain complexes that there is freedom. Also, it is in the nature of things that the in nite cyclic group Z should have a prominent place aside from its role in the UCT; in the absence of other constraints we can most conveniently form chain and cochain complexes from free abelian groups. In the sequel we shall see that sometimes a problem can be solved by means of a homology theory simpli ed by using chain complexes of modules over Z2, instead of free abelian groups which are modules over Z. In other cases a considerable gain is made by using chain complexes of modules over R, which are simply real vector spaces, for that permits the use of powerful theorems from linear algebra. Usually a homology theory arises from natural classes of entities, typically from classes of simplices with common dimension. Correspondingly, a cohomology theory often arises from homomorphisms de ned on free abelian groups, or modules. We shall see several such cases in the ensuing examples. Actually, geometric simplicial complexes have wider application than might at rst be anticipated. For, if we distill out their essential algebra that is needed to construct a chain complex, then we obtain an abstraction which admits application when no geometric interpretation is available. This is exempli ed by C ech homology and cohomology where the basic building blocks are abstract simplices 143

144

CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

whose `vertices' are sets from an open cover. In order that these essential algebraic ingredients of simplices can be identi ed in particular cases we make:

De nition 5.0.1 An abstract simplicial complex K is a pair (V; S ), where V is a set of elements called vertices of K and S is a collection of nite nonempty subsets of V called simplices of K , satisfying S (i) f j  2 S g = V (covering property), (ii) ? = 6   2 S ) 2 S (face-saving property). If  2 S contains (m + 1) vertices then we say  is an m-simplex. Ex on abstract simplicial complexes 1. If (V; S ) is an abstract simplicial complex with V a nite set, then it always can be realized as a geometric simplicial complex (cf. Hocking and Young [46], p. 213). 2. If X is a space in Top, for example a compact Hausdor space, then every open cover U of X has a (locally) nite subcover VU = fU1; U2 ; : : : ; Uk g which de nes an abstract simplicial complex (VU ; SU ) where

 2 SU $  = fUki j i = 0; 1; 2; : : :; mg T U 6= ?, for some m  0. with m i=0 ki In the next few sections we summarize some common theories and give some notes on their construction. These summaries give the simplest case for each theory, namely the absolute one. In the sequel, we shall encounter generalized theories arising from spectra.

5.1 Cubical singular homology Generators: Continuous maps c : Im ! X generate Qm, for m  0. Boundary operation: @c =

m X

( 1)i (0i  c 1i  c)

i=1 with ri  c(x) = c(x1 ;


; xi 1 ; r; xi+1 ;


; xm 1 ).

Chain group:

Cm (X ) = Qm =Dm where Dm is the set of c for which c(x) is independent of coordinate xk for some k  m. Morphisms: Continuous f : X ! Y give f via f  c.

5.1. CUBICAL SINGULAR HOMOLOGY

145

De nition 5.1.1 A singular m-cube (for a non-negative integer m) in a topological space X is a continuous map c : Im ! X : (x1 ; x2 ; : : : ; xm ) 7! c(x1 ; : : : ; xm ) where I0 = f0g and Im = [0; 1]m  Rm . Such a c is called degenerate if there exists k with 1  k  m such that c(x1 ; x2 ; : : : ; xm ) does not depend on xk ; we

throw these away algebraically. Denote by Qm (X ) the free abelian group generated by the set of all singular m-cubes in X and let Dm (X ) be the subgroup generated by degenerate m-cubes in X . We de ne the quotient group, Qm (X )=Dm(X ) = Cm (X ), to be the group of cubical singular m-chains in X , with Cm = 0 for m < 0.

We can easily make Cm (X ) a module over any ring R and so, to obtain a chain complex, we need to construct a boundary operator @m . This is achieved in terms of the following face operators.

De nition 5.1.2 For each k = 1; 2; : : :; m > 0 and q = 0; 1 we de ne qk : Qm (X ) ! Qm 1 (X ) : c 7 ! qk  c by putting

qk  c(x1 ; : : : ; xm 1 ) = c(x1 ; : : : ; xk 1 ; q; xk+1 ; : : : ; xm 1 ) : We call 0k  c the front k-face of c and 1k  c the back k-face of c.

Theorem 5.1.3 (Cubical boundary operator) For each m > 0 there is a group

homomorphism

@~m : Qm (X ) ! Qm 1 (X ) : c 7 !

m X k=1

( 1)k (0k  c 1k  c) :



Corollary 5.1.4 Since @~m preserves degeneracy, it follows that it induces a group

homomorphism on the quotients and we obtain

@m : Cm (X ) ! Cm 1 (X ) and @m 1 @m = 0.



Theorem 5.1.5 (Cubical singular chain complex) For any Top space X , fCm (X ); @m j m 2 Zg with Cm (X ) = 0 for m < 0 ; is a chain complex of abelian groups called the cubical singular chain complex of X . Every continuous map f : X ! Y determines a unique homomorphism Qm (X ) ! Qm (Y ) : c 7 ! f  c

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

for each m = 0; 1; 2; : : :. Since this preserves degeneracy, it determines a chain map

f] : fCn (X )g ! fCn (Y )g and hence also homomorphisms of quotients m f = fHm (X ) f! Hm (Y ) j m 2 Zg where Hm (X ) = imker@ @m : m+1

There is a covariant functor of degree 0 into the category of graded abelian groups

fHm (X ) j m 2 Zg

X H : Top /



GrAb : f

H (f ) /

fHm(Y ) j m 2 Zg

Y





which gives rise to cubical singular homology theory (H ; @ ) on Top, with @ induced by the @m . 

There is, of course, considerable work in proving this by checking the axioms. We turn to extending our theory to Topp.

Theorem 5.1.6 (Relative cubical singular homology) There is a covariant

functor, also denoted by H ,

(X; A)

H : Topp /

GrAb :



f

(Y; B ) 

fHm (X; A) j m 2 Zg H (f ) /

fHm (Y; B ) j m 2 Zg 

agreeing with the previous one on Top.

Proof: We outline the steps. i (i) Given (X; A) with A ,! X in Top, then i] : Cm (A) ! Cm (X ) is well de ned and monic so Cm (X; A) = Cm (X )=Cm (A) is well de ned in Ab.

(ii) Since @m Cm (A)  Cm 1 (A), it extends to a homomorphism

@m : Cm (X; A) ! Cm 1 (X; A) and fCm (X; A); @m j m 2 Zg is a chain complex of abelian groups. (iii) De ne H0 (X; A) = C0 (X; A)= im @1 , Hm (X; A) = ker @m = im @m+1 for m 6= 0.

5.2. SIMPLICIAL SINGULAR HOMOLOGY

147

(iv) Given f : (X; A) ! (Y; B ) in Topp then we have

f] = fCm (X ) ! Cm (Y ) j m 2 Zg and, since fA  B , (f jA )] = fCm (A) ! Cm (B ) j m 2 Zg as chain maps. Hence there is an extension to the chain of quotients

fCm(X; A) ! Cm (Y; B ) j m 2 Zg and therefore a well-de ned homomorphism in homology

f = fHm (X; A) ! Hm (Y; B )jm 2 Zg :



From the foregoing construction we obtain on Topp the relative cubical singular homology theory, usually also denoted (H ; @). We observe that, from the construction of the chain complexes, we always have

Hm (X ) = 0 = Hm (X; A) for m < 0 : Again,the notation

Zm(X; A) = ker @m Bm (X; A) = im @m+1

is standard and conveniently keeps the underlying spaces in view.

5.2 Simplicial singular homology Generators: Continuous maps  :
m ! X , where
m is the standard m-simplex, generate Am . Boundary operation: @ = j@
m Chain group: Sm (X ) = Am ; m  0 ; Sm(X ) = 0; m < 0 : Morphisms: Continuous f : X ! Y give f via f  . De nition 5.2.1 The standard m-simplex
m in Rm+1 is the set he0 ; : : : ; em i where e0 = (1; 0; : : :; 0); e1 = (0; 1; 0; : : :; 0); e2 = (0; 0; 1; : : :; 0); : : : : Thus
m is a closed, convex m-dimensional subset of Rn .

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

A singular m-simplex in a topological space X is a continuous map  :
m ! X : We denote by (e0 ; : : : ; em ) its image and by Sm (X ) the free abelian group generated by all the singular m-simplices in X , with Sm (X ) = 0 for m < 0. Then @m : Sm (X ) ! Sm 1 (X ), the boundary operator, is the homomorphism de ned on generators by

@m : (e0 ; : : : ; em ) 7 !

m X

( 1)r (e0 ; : : : ; e^r ; : : : ; em )

r=0 where e^r denotes omission of vertex er . [Those familiar with the exterior derivative

should recognize this pattern.]

Theorem 5.2.2 (Simplicial singular chain complex) The set S (X ) = fSm (X ); @m j m 2 Zg is a chain complex of abelian groups, called the (simplicial) singular chain complex of X .  We can now construct a functor Top ! GrAb by sending spaces to the homology groups of these chain complexes and by sending continuous maps f : X ! Y to homomorphisms f : fHm(X )g ! fHm(Y )g arising from the chain maps

fm : Sm (X ) /


m 

Sm (Y ) :  X


m f  /

Y Then, as for the cubical case, we can extend the functor to Topp by considering the 

chain complex of quotients

Sm (X; A) = Sm (X )=Sm(A) :



In this way we obtain (simplicial) singular homology and relative (simplicial) homology. As might be suspected no new information is gained; we obtain groups that are isomorphic to those for corresponding cases in cubical theory. The latter theory is, however, more convenient when relating homology groups and higher homotopy groups. We leave the proofs of the next few results as exercises.

Theorem 5.2.3 (Homology of contractible spaces)  n = 0; Hn () = Z0 for otherwise. Thus this is an ordinary theory with coecients Z.

5.2. SIMPLICIAL SINGULAR HOMOLOGY

149

Ex Calculate H(Sn) again. Clearly a singleton space is topologically trivial, but is not a trivial set: it is nonempty. Thus it is H0 which is the obstruction to vacuity, entirely proper from the algebraic point of view. However, from the point of view of applications, singletons may create only pathological interest. Thus we may conveniently factor out from homology this indicator of nonemptiness, obtaining reduced homology H~  : H~ n = Hn except H~ 0  Z = H0 : In order to achieve this as a homology theory (H~  ; @~) in its own right, we introduce the reduced chain complex S~: S~n = Sn except S~0  Z = S0 : Theorem 5.2.4 (Reduced homology) Using the appropriate de nition for @~, (H~  ; @~) is a homology theory. 

Theorem 5.2.5 (Long exact homology sequence) Given (X; A) f! (Y; B )

in Topp with induced

j S (A) ,!i S (X ) ! S (X; A)

in C hain, there is a short exact sequence

i j 0 ! S (A) ,! S (X ) ! S (X; A) ! 0 : Then the exactness theorem for chain complexes yields the exact sequence in homology and there is the following commutative diagram with exact rows




/

Hn (A) i Hn (X ) /

f






Hn (X; A) @ Hn 1 (A) /

Hn (B ) i0

Hn (Y )

f



/

f

Hn (Y; B ) @0 Hn 1 (B ) 

j0 /

/






/

f 

/

j



/

There is a corresponding diagram for reduced homology.

/






Corollary 5.2.6 Given any x 2 X , then X ,! (X; fxg) determines an exact homology sequence. Since H~ n (fxg) = 0 for all n, it follows that H~ n (X )  = Hn (X; fxg) for all n ; because the exact sequence breaks up into pieces like 0 ! H~ n (fxg) ! Hn (X; fxg) ! 0 :



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5.3 Cup product We shall now construct the cup product. As mentioned earlier, the presence of products in cohomology provides an additional source of information, helping us to distinguish between otherwise similar spaces (page 130). First, we need to be able to view the (singular) faces of a singular m-simplex. De ne the inclusion opposite the ith vertex, 1  i  m + 1, i :
m ,!
m+1 : (x0 ; : : : ; xm ) 7 ! (x0 ; : : : ; 0; : : : ; xm ) where the new 0 occurs in the ith coordinate position. For convenience, we shall indicate k-fold composition via i1


ik = i1  i2 


 ik : Now, for a singular m-simplex  :
p+q ! X , de ne the front p-face p  =   p+1


p+q and the back q-face q =   0


p : These may now be used together with the Kronecker product (or pairing) to de ne the internal or cup product of a p-cocycle u and a q-cocycle v: huv; i = hu [ v; i = ( 1)pq hu; p ihv; q i : This makes sense for integer coecients, of course, but also more generally whenever the coecients are a ring R regarded as a Z-module.

Ex on cup product in cohomology

1. The product on cochains passes to cohomology to de ne the internal or cup product on cohomology, that is (u [ v) = u [ v + ( 1)p u [ v : 2. Show H~  (Sn; R) is a trivial ring for all n  1. More generally, this holds for any suspension SX . 3. Let p be a prime and consider the coecient ses 0 ! Zp ! Zp2 ! Zp ! 0 with Bockstein (cf. page 111). Then = 0 and (u [ v) = (u) [ v + ( 1)nu [ (v) for u; v 2 H n ( ; Zp). When p = 2, then : H 1 ( ; Zp) ! H 2 ( ; Zp) : u 7! u2 . 4. Try some calculations with coecients in Z2. See also section 5.11. Henceforth, we shall normally consider only cohomology with coecients in a ring (or sheaf of rings). Geometrically, this cohomology product corresponds roughly to the product of cells. In particular cases, the correspondence may be quite precise.

5.4. GEOMETRIC SIMPLICIAL HOMOLOGY

151

Ex on cohomology of simple products 1. H (Sn  Sm; Z) has generators (as a ring) 1, the 0-cell, en, the n-cell, em , the m-cell, and the product en [ em = en em , the (n + m)-cell which is the product of the n-cell and the m-cell. Thus

H k (Sn  Sm; Z)  =

 Z ; k = 0; n; m; n + m;

0 otherwise. If n = m, we naturally get Z  Z at k = n = m. Compare this with the trivial ring structure for Sn _ Sm _ Sn+m. 2. Investigate H  (C P n ; Z) and H (RP n ; Z2).

5.4 Geometric simplicial homology We de ned geometric simplicial complexes in the previous chapter and saw how useful they could be for computing fundamental groups of triangulable spaces. They give also an intuitively appealing homology theory which re ects well its role as a `hole-ology' theory. For this we need to use oriented simplices so that we can incorporate them in an abelian group structure. An orientation for a simplex is just a choice of ordering of its vertices; two choices are called equivalent orientations if they dier by an even permutation of the vertices. It follows that there are just two equivalence classes under this relation for simplices of dimension greater than zero; so there are two choices for the orientation of such simplices.

Ex The choice of an orientation for a simplex  induces an orientation on each of its faces and @ ( ) = @; check this for  =
3 . Generators: Oriented m-simplices  = hv0 ; : : : ; vm i in a simplicial complex K yield a free abelian group Am . Boundary operation: @ =

Chain group:

m X i=0

( 1)i hv0 ; : : : ; v^i ; : : : ; vm i

Sm (X ) = Am ; m  0; Sm (X ) = 0; m  0 : Morphisms: Continuous f : jK j ! jLj between polyhedra give f via the Sim-

plicial Approximation Theorem. Recall that a geometric simplicial complex K is a locally nite set of (henceforth, oriented) simplices, all contained in some Rn , satisfying the conditions:

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(i) if k is a face of m 2 K , then k 2 K ; (ii) if k ; m 2 K then k \ m is empty or it is a common face of k and m . Note that our simplices are closed; if we had used open ones then we would require

k \ m 6= ? () k = m : A face of an m-simplex m is the subspace simplex generated by any subset of the vertices of m . The dimension of an m-simplex is m, in agreement with ane geometry. The dimension of K is the maximum of the dimensions of its simplices. A subcomplex of K is a subset L  K satisfying (i), and hence (ii). If m 2 K , then the set m of faces of m is a simplicial complex and the set _ m of proper faces of m is a simplicial complex. For each r  0, the r-skeleton K r is the subcomplex of K consisting of all oriented simplices of dimension at most r. The star stK () of a simplex in K is the union of the interiors of the simplices of K that have  as a face. A simplicial pair (K; L) is a simplicial complex K and one of its subcomplexes L. The polyhedron or realization of K , denoted by jK j, is the topological subspace of Rm consisting of the points of simplices of K .

De nition 5.4.1 Given simplicial complexes K and L, a simplicial map is any map f : jK j ! jLj satisfying: (i) f sends vertices to vertices; (ii) ha0 ; : : : am i 2 K ) ff (a0 ); : : : ; f (am )g spans a simplex of L, so f sends simplices to simplices; (iii) f is linear on each simplex, in the sense that

x=

m X i=0

i ai 2 ha0 ; : : : ; am i 2 K =) f (x) =

m X i=0

i f (ai ) :

A simplicial pair map

f : (jK j; jLj) ! (jM j; jN j) is a simplicial map

f : jK j ! jM j with f (jLj)  jN j : It turns out that conditions (i){(iii) ensure that a simplicial map must always be continuous. We can easily check that there is a category of geometric simplicial complexes and simplicial maps and a corresponding one for simplicial pairs. The whole theory can be detached from dependence on Rn by using, instead of geometric simplicial complexes, abstract simplicial complexes. These consist of a set V , and a set S of locally nite non-empty subsets of V which covers V and which is closed

5.4. GEOMETRIC SIMPLICIAL HOMOLOGY

153

under the taking of proper subsets (i.e., under the formation of faces as de ned on page 144) The important thing is that any locally nite abstract simplicial complex (i.e., the vertex set is locally nite) has a geometric realization (e.g., in R2q+1 if it has q + 1 vertices) and any two realizations of the same complex are isomorphic in the category of geometrical simplicial complexes.

De nition 5.4.2 Let (K; L) be a simplicial pair. Sn(jK j) is, from above, the free abelian group generated by continuous maps  :
m ! jK j and we denote by
m (K ) that subgroup generated by simplicial . De ne:


m (K; L) =
m (K )=
m(L) and observe that it may be viewed as a subgroup of Sm (jK j; jLj).

Proposition 5.4.3 We have two subchain complexes, of S (jK j) and S (K; L), re
(K ) = f
m (K ); @m j m 2 Zg
(K; L) = f
m (K; L); @m j m 2 Zg : We denote by H (K ) and H (K; L) the simplicial homology of the chain comspectively:

plexes
(K ) and
(K; L) respectively. The signi cant result is that for all m there are isomorphisms with the singular homology groups (simplicial or cubical) of the corresponding polyhedra:

Hm (K )  = Hm (jK j) Hm (K; L) = Hm (jK j; jLj) :

These are constructed via a chain map which is an inverse (up to chain homotopy) of the inclusion chain map
(K ) ,! S (jK j). Singular m-simplices in Sm (jK j) are approximated by simplicial maps of triangulations of
m ; the theorem at work is the Simplicial Approximation Theorem. The homology theory that arises is (oriented) simplicial homology and a reduced homology theory is obtained as before.

De nition 5.4.4 For simplicial complexes K; L and a continuous map f : jK j ! jLj, we say that a simplicial map g : jK j ! jLj is a simplicial approximation to f if for every vertex v 2 K we have f (stK v) = stL (g(v)) : Then it is easy to establish the following.

Theorem 5.4.5 (Approximation related to composition and homotopy)

If g is a simplicial approximation to f , then g is homotopic to f . Simplicial approximation is preserved under composition of maps. 

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

Not every continuous map f : jK j ! jLj has a simplicial approximation, because there may be too few vertices in K to accommodate the variability of f in a simplicial way. However, if we are prepared to introduce more vertices in the interiors of existing simplices of K then we can always nd a simplicial approximation. A systematic procedure for introducing extra vertices in K is to place a new vertex at the barycenter (literally, center of gravity) of each simplex in K and join it to the original vertices of that simplex. The resulting complex K 0 is called the ( rst) barycentric subdivision of K or the derived complex of K . In a particular case such a complete subdivision might be unnecessary and it might be sucient to leave a whole subcomplex, K0 say, of K unaltered during the subdivision; then we have stellar subdivision relative to K0 . In any case we can repeat the procedure inde nitely, so obtaining K (r) the rth subdivision for any r 2 N . This leads to the important theorem that we have met already on page 89.

Theorem 5.4.6 (Simplicial approximation) Given simplicial complexes K; L and continuous f : jK j ! jLj, then for some r 2 N there is a simplicial approximation to f : jK (r) j ! jLj.  Corollary 5.4.7 The theorem extends to simplicial pairs.



Ex 1. Every simplicial complex K with m components has H0 (K ; Z) isomorphic to m copies of Z. 2. Construct an example of composite simplicial maps

@
3 !
2 !
1 and calculate the induced chain maps and homomorphisms. 3. Two simplicial approximations a; b to a map

f : jK j ! jLj are close in the sense that for all x 2 jK j the carrier of f (x) contains a(x) and b(x). Close simplicial maps induce the same homomorphisms in homology and yield a chain homotopy

dk : Ck (K ) ! Ck+1 (L) with dk 1 @ @dk = ak bk : Ck (K ) ! Ck (L) for all k. Details of the construction of the chain maps leading to the isomorphisms between singular and simplicial homology groups can be found in, e.g., Maunder [68].

5.5. COMPUTING SIMPLICIAL HOMOLOGY GROUPS

155

5.5 Computing simplicial homology groups The representation of a space X as a simplicial complex K allows the de nition of its simplicial homology which turns out to be independent of the choice of triangulation K and gives, moreover, a homotopy-type invariant functor. The steps in establishing this are summarized as follows.

Ex on simplicial homology invariance 1. Every continuous f : jK j ! jLj induces by SAT a simplicial approximation, for some barycentric subdivision K m of K , s : jK mj ! jLj with f  s :

2. Simplicial maps induce chain maps by sending a simplex to its image simplex if it identi es no two vertices, and otherwise to 0. Chain maps which commute with the boundary operators yield homomorphisms of homology groups. Compositions of chain maps carry through to their composites in homology. 3. By using stellar subdivision (cf. page 154 and Armstrong [2], p. 186) we can simplify barycentric subdivision into smaller steps as subdivision of one selected simplex in a complex. Stellar subdivision induces an isomorphism on homology and it follows that barycentric subdivision does not alter the simplicial homology of a complex. 4. By composing the barycentric subdivision chain map with the one induced through simplicial approximation, we obtain an induced morphism in homology for each continuous map between polyhedra. This induction is independent of the choice of simplicial approximation because if two simplicial approximations are taken then they always send a simplex to faces of one simplex in the image complex. 5. Composition of continuous maps between polyhedra is preserved in homology, and homotopic maps yield identical homomorphisms. We can see the kind of steps involved in computing simplicial homology groups by the case of S1 triangulated as the boundary of the standard 2-simplex with vertices v0 ; v1 ; v2 . The various groups are as follows: C0 = hv0 ; v1 ; v2 i so @0 C0 = 0 C1 = hv0 v1 ; v1 v2 ; v2 v0 i Ck = 0 ; k 6= 0; 1 ker @0 = hv0 ; v1 ; v2 i im @1 = hv1 v0 ; v2 v1 ; v0 v2 i H0 = hv0 ; v1 ; v2 i=hv1 v0 ; v2 v1 ; v0 v2 i  =Z ker @1 = h(v0 v1 + v1 v2 + v2 v0 )i  =Z im @2 = 0 H1 = h(v0 v1 + v1 v2 + v2 v0 )i=0  = Z:

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

A similar but longer computation using the boundary of the standard (n+1)-simplex yields for n > 0

H0 (Sn)  = Hn (Sn)  = Z ; otherwise Hk (Sn) = 0 :

For orientable closed surfaces, the sum of all the 2-simplices generates the 2dimensional homology. However, this sum is not a 2-cycle in the nonorientable case and here the 2-dimensional homology is trivial. We assemble some aids to computation. Recall that for any group G its commutator subgroup is generated by

fx 1 y 1 xy j x; y 2 Gg :

In fact, the commutator subgroup is the smallest normal subgroup C such that G=C is abelian. The transition from G to G=C is called abelianization of G.

Ex Distinguish between the free product Z  Z and the direct sum Z  Z and show that there is an exact sequence

0 ! C ! Z Z ! Z Z ! 0 in which C is the commutator subgroup of Z  Z.

Theorem 5.5.1 (Abelianizing fundamental groups) If K is a connected simplicial complex with fundamental group 1 (jK j) having commutator subgroup C then H1 (K )  = 1 (jK j)=C :

Proof: Take any vertex v 2 K then 1 (jK j)  = E (K; v); the edge group at v: Every edge loop determines a 1-cycle of ordered pairs of vertices and homotopic edge loops determine homologous 1-cycles. Hence we get a map

 : E (K; v) ! H1 (K ) : This turns out to be a homomorphism with kernel the commutator subgroup of

1 (jK j) and so the result follows.

Ex On simplicial homology 1. Fill in the details to show that  is an isomorphism. 2. Find H1 (S1 _ S1). 3. Find H1 (X ) where X is (a) a sphere with a disk replaced by a Mobius strip;



5.6. RELATIVE SIMPLICIAL HOMOLOGY 4. 5. 6.

7.

8.

157

(b) a sphere with one handle. If K is an orientable closed surface then the sum of all 2-simplices generates H2 (K ). If K is a nonorientable closed surface then H2 (K ) = 0. Given a connected space X , a cut point of X is any x 2 X such that X nfxg is not connected. Cut points are topological invariants but not continuous invariants. Show that S2 _ S1 _ S1 has the same homology as S1  S1 but these two spaces are not homeomorphic. Since simplicial homology groups of polyhedra are nitely generated abelian groups, they always admit a decomposition into a direct sum of a number of copies of Z and a torsion part containing the nite order elements. The number of copies of Z in Hk (K ) is the kth Betti number of K . Betti numbers are homotopy type invariants and so spaces with diering Betti numbers cannot be of the same homotopy type, nor therefore homeomorphic. Consider the homology H (K ; Z2) of an n-dimensional simplicial complex K , using coecients from Z2. Denote by bk the number of copies of Z2 in Hk (K ; Z2). Then n X (K ) = ( 1)kbk : i=0

Theorem 5.5.2 (Classi cation of closed surfaces) If K is a simplicial complex triangulating a closed surface X , then its rst homology group is given by the following classi cation:  is orientable of genus h ; H1 (K )  = h(bZ 1)Z  Z2 ifif X X is nonorientable of genus b : Proof: This depends on the technique of surgery to show that all closed surfaces are homeomorphs of standard ones arrived at by adding handles or Mobius strips in place of disks on the surface of a sphere. An excellently detailed account of this with illustrative diagrams is given in Armstrong [2], who goes on to compute nite presentations of the fundamental groups for all closed surfaces. Their abelianizations turn out to be sucient to distinguish all types.  The theorem of Rado [91] established that closed surfaces, that is compact 2manifolds without boundary, always admit triangulations so the result is stronger than it rst appears.

5.6 Relative simplicial homology

Proposition 5.6.1 For any m-simplex m, its set of faces m and its set of proper faces _ m form a simplicial pair and  m; Hk (m ; _ m ) = 0Z;; kk 6= = m:

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

Proof:
k (m ) =
k (_ m ) for k 6= m so the quotients are trivial, but
m (m )=
m (_ m ) is non trivial, with one generator [@m ].



De nition 5.6.2 Given two subcomplexes K1; K2 of a simplicial complex K , their Mayer-Vietoris sequence is the exact sequence of simplicial homology groups  H (K [ K ) @!  H (K \ K ) i!  H (K )  H (K ) j! 





@! k 1 2 k 1 2 k 1 k 2

induced by the short exact sequence of chain complexes j i
(K ) 
(K ) ! 0 !
(K1 \ K2 ) !
(K1 [ K2 ) ! 0 : 1 2

De nition 5.6.3 The relative Mayer-Vietoris sequence for simplicial pairs (K1 ; L1 ); (K2 ; L2) in K is the exact sequence.  H (K L )  H (K L )


! Hk (K1 \ K2 ; L1 \ L2 ) i! k 1 1 k 2 2

j

 H (K \ K ; L \ L ) !


! Hk (K1 [ K2 ; L1 [ L2) @! k 1 1 2 1 2

induced by the short exact sequence

0 !
(K1 \ K2 )=
(L1 \ L2 ) ! (
(K1 )=
(L1 ))  (
(K2 )=
(L2 ))

!
(K1 [ K2 )=
(L1 [ L2 ) ! 0 : In singular homology, we have to restrict the choice of subsets to account for the loss of simplicial information. Given a topological space X then a pair (X1 ; X2 ) of its subspaces is called an excisive pair if the excision map e (X1 ; X1 \ X2 ) ,! (X1 [ X2 ; X2 )

induces an isomorphism in singular homology. A sucient condition for this to happen, as anticipated by the excision axiom, is that X2  X1 ; then e is the identity. For any X1 ; X2  X we always have a short exact sequence of singular chain complexes j i S (X )  S (X ) ! 0 ! S (X1 \ X2 ) ! S (X1 ) + S (X2 ) ! 0 1 2

where S (X1 ) + S (X2 ) is generated by S (X1 ) [ S (X2 ), because

S (X1 \ X2 ) = S (X1 ) \ S (X2 ) : However, in the case that (X1 ; X2 ) is an excisive pair we have the isomorphisms Hk (S (X1 ) + S (X2 ))  = Hk (S (X1 [ X2 )) :

5.6. RELATIVE SIMPLICIAL HOMOLOGY

159

Ex Simplicial pairs are automatically excisive. De nition 5.6.4 The singular Mayer-Vietoris sequence of an excisive pair (X1 ; X2 ) is the exact sequence

 H (X )  H (X )  H (X \ X ) i! : : : @! k 1 k 2 k 1 2

j

! Hk (X1 [ X2 ) ! Hk 1 (X1 \ X2 ) ! : : :

For comparison, we calculate the homology of spheres again (cf. Corollary 4.3.11 and the Ex on p. 115).

Theorem 5.6.5 (Homology of spheres) The homology groups of spheres are: 8 Z Z; m = n = 0; < Hm (Sn)  = : Z ; m = n > 0; or m = 0; n > 0; 0

otherwise.

Proof: Sn is homeomorphic to _ where  is the standard (n +1)-simplex. Now,  is a simplicial complex and j j is contractible. Consider the reduced chain complexes ~ _ ) and
( ~  ). Since j j is contractible, its reduced homology is trivial. Also,
( ~
m (_ ) = 0 if m > n or if m  1. There is an inclusion i : j_ j ,! j j which is evidently simplicial and induces a chain map i with
~ m (_ )  =
~ m ( ) for all m  n:


0 0
~ n (_ ) @
~ n 1 (_ )
~ 1 (_ ) /

/

/


~ n ( ) 

/

/

in 1

in


~ n+1 () @ We deduce that

/


~ n 1 ( ) 

i1

@ /

/







~ 1 ( ) 

/

/

0

H~ m (_ )  = H~ m ( ) = 0 for m  n :

Next: H~ n (_ )  = Z~n (_ ) since B~n (_ ) = 0;  ~ Zn(_ ) = Z~n ( ) since in is an isomorphism; Z~n( )  = B~n ( ) since H~ n () = 0. ~ Now, Bn () has only one generator, @ [], hence so does H~ n (_ ) and the result follows.



Simplicial Homology with Rational Coecients

For an oriented simplicial complex K we de ned homology groups from quotients of free abelian groups of oriented k-simplices. Now, a free abelian group is a module over Z; intuitively, a module is like a vector space over a ring instead of a eld. Here, our basis elements are the generators of the group. In fact, we could replace

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Z by any abelian group without problems. For the present purposes we shall replace Z by the rational eld Q , so that we can exploit linear algebra in the context of

homotopy type invariants for the next series of results. The formal steps are quite easily comprehended and we just summarize them as follows. Let k denote the number of k-simplices in K . For each k we construct the vector space Ck (K ; Q ) over Q , with basis consisting of oriented k-simplices. The vector space operations are induced directly from Q . Then, dim Ck (K ; Q ) = k . The collection of these vector spaces becomes a chain complex with boundary operators

@kQ : Ck (K ; Q ) ! Ck 1 (K ; Q )

induced by linearity over Q from those de ned over Z.

Theorem 5.6.6 (Rational simplicial homology) Let K be a ( nite) simplicial complex with k , k-simplices for k = 0; 1; : : :; n. The vector space quotient Q Hk (K ; Q ) = ker @Qk im @k+1

of subspaces of Ck (K ; Q ) has dimension equal to the rank of the free abelian part of H (K ), the kth Betti number k of K .

Proof: First consider homology with coecients in Z. Since H (K ) is a nitelygenerated abelian group, it admits a decomposition into a free part Fk and a torsion part Tk . Let a minimal set of generators of Fk be denoted by f[z1 ]; : : : ; [z k ]g and of Tk by f[w1 ]; : : : ; [w k ]g, so the [wi ] all have nite order. Let z be a k-cycle, so @z = 0 and [z ] 2 Hk (K ). Denoting by hz i the corresponding element of Hk (K ; Q ), we shall show that fhz1 i; : : : ; hz k i; hw1 i; : : : ; hw k ig

spans Hk (K ; Q ). Suppose that we have a rational k-cycle r a X i

i 2 ker @kQ  Ck (K ; Q ) b i i=1

for some integers ai ; bi with bi 6= 0. Then we can rearrange it as: r X 1 ci i  = i b b


b 1 2 r i=1 i=1 bi

r a X i

for some ci 2 Z, and with

r X i=1

ci i 2 ker @k  Ck (K ; Z) :

Now, ker @k is spanned by z1 ; : : : ; z k ; w1 ; : : : ; w k , so

fhz1 i; : : : ; hz k i; hw1 i; : : : ; hw k ig

5.6. RELATIVE SIMPLICIAL HOMOLOGY

161

spans Hk (K ; Q ). Next we get rid of all the nite-order elements from our spanning set. Suppose that [w] 2 Hk (K ) has nite order m. Then we can nd

h 2 Ck+1 (K ; Z) with @k+1 h = mw; so mw 2 im @k+1 : By construction of @ Q, it follows that

1 

@k+1 m h = w 2 Ck (K ; Q ) so hwi = 0 ; Q

and the result follows. Finally, we have to show linear independence of the set fhz1i; : : : ; hz k ig  Hk (K ; Q ). Suppose that we have r a X i i=1 bi

zi = 0 2 Hk (K ; Q )

for some integers ai ; bi . Then r a X i

zi = @kQ+1 h; for some h 2 Ck+1 (K ; Q ) : b i i=1

However, clearing the denominators,

b1 b2


br So we have and

b1 b2


br r X

P

i=1

r a X i

r a X i i=1 bi

zi 2 Ck (K ; Z) :

r X

zi = ci zi ; for some ci 2 Z ; i=1 i=1 bi

ci zi = @k+1 g; for some g 2 Ck+1 (K ; Z) :

Then ri=1 ci [zi ] = 0 but the [zi ] are linearly independent because the zi constituted a minimal set of generators for F . Hence, ci = 0 for all i and so dim Hk (K ; Q ) = k :



This construction allows us to prove that the classical Euler characteristic of a nite simplicial complex is independent of the triangulation and actually a homotopytype invariant dependent only on the Betti numbers, coinciding with our earlier version in De nition 4.3.23.

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Theorem 5.6.7 (Euler-Poincare) If K is a nite simplicial complex with k , ksimplices and kth Betti number k , then the Euler characteristic (number) is given by (K ) =

dim XK

k=0

( 1)k k =

dim XK

k=0

( 1)k k :

Proof: This is an exercise in linear algebra, exploiting the subspace inclusions im @kQ+1 = Bk (K ; Q )  ker @kQ = Zk (K ; Q )  dom @kQ = Ck (K ; Q ) : As almost always in linear algebra proofs, we repeatedly apply the process of extending a base from a subspace to a larger space, and employ the rank and nullity theorem for a linear map, in our case @kQ, as dim dom @kQ = dim ker @kQ + dim im @kQ : Letting k = dim im @kQ, we nd that

k = ( k+1 + k ) + k with 0 = n+1 = 0. Then the k 's cancel in the alternating sum and the result follows. 

Corollary 5.6.8 The classical Euler characteristic is a homotopy type invariant for compact triangulable spaces, because the Betti numbers are. Therefore:

(K ) 6= (K 0 ) =) jK j 6 jK 0j =) jK j  6 jK 0 j : =



Ex 1. Find a cover for S2 consisting of 3 closed sets and show that one of them must contain an antipodal pair. 2. Supply the linear algebra details to establish the Euler-Poincare formula. [Hint: Start at the top dimension, extending from Bn (K ; Q ) through bases for Zk (K ; Q ) and Ck (K ; Q ).] 3. Compute the Euler characteristic for some low dimensional polyhedra and deduce bounds on the sizes of their homology groups. Next we show that we do not need to go to homology classes to obtain the trace in the formula for the Lefschetz number; we can work at the chain group level.

Theorem 5.6.9 (Hopf trace) Let X be a compact triangulable space with triangulation jK j, let C (K ; Q ) be the simplicial chain complex with rational coecients, and let  : C (K ; Q ) ! C (K ; Q )

5.6. RELATIVE SIMPLICIAL HOMOLOGY

163

be a chain map. Then  induces for each k = 0; 1; : : :; n = dim K a linear map

k : Ck (K ; Q ) ! Ck (K ; Q ) which factors through to homology to give

k : Hk (K ; Q ) ! Hk (K ; Q ) with

n X

n X

k=0

k=0

( 1)k tr k =

( 1)k tr k :

Proof: Choose `standard' bases for Ck (K ; Q ) as in the proof of the Euler-Poincare formula and use the fact that, by de nition of a chain map, it commutes with the boundary operator.  Now we give a proof of the principal part of the Lefschetz Fixed Point Theorem for simplicial complexes; cf. Theorem 4.3.41 for the complete statement.

Theorem 5.6.10 (Lefschetz xed point) Let f : X ! X be a continuous map from a compact triangulable space to itself and let f be the Lefschetz number of f . If f 6= 0, then f has a xed point. Proof: Suppose that f is given and a triangulation homeomorphism h : jK j ! X is chosen. This induces a map

f h : jK j ! jK j : Assume to get a contradiction that f has no xed point; then f h has no xed point. Now construct g : jK j ! R : x 7 ! d(x; f h (x)) which measures the distance that f h moves each point. This g is clearly continuous and never zero; it attains a lower bound  > 0, say. By using if necessary a barycentric subdivision, arrange that the longest edge in K has length less than =3. By the Simplicial Approximation Theorem, we can nd a simplicial approximation to f h , say s : jK m j ! jK j ; and a subdivision chain map for K m, say  : C (K ; Q ) ! C (K m ; Q ) :  7 !

X j

j :

The induced map on homology is:

fkh = sk  k : Hk (K ; Q ) ! Hk (K ; Q ) :

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By the Hopf Trace Theorem, we need only consider the trace of

sk k : Ck (K ; Q ) ! Ck (K ; Q ) : Take any (oriented) simplex i 2 Ck (K ; Q ) and nd  2 k (i ), where, of course, k (i ) is just a linear combination of the simplices in i . Now, since s approximates f h , for all x 2 jK j, d(f h (x); s(x)) < =3 and so both points lie inside the same simplex in K m. There are two cases to consider: either sk ( )  sk k (i ) is sent to i , whereupon for x 2  , we get a contradiction by d(x; s(x)) < =3 ) d(x; f h (x)) < =3 + =3 <  : or sk ( ) = 0 for all  2 i , whereupon

sk k (i ) =

X j

j j has all j = 0

P

and then the trace of sk k is simply j j = 0. We conclude that f h, and hence also f , must have a xed point.



Corollary 5.6.11 1jKj = (K ).



Ex 1. Write out the matrix algebra needed in the proof of the Hopf Trace Theorem. 2. Write out the matrix algebra needed in the proof that f is well-de ned. Illustrate it explicitly for the antipodal map on S2.

5.7 Geometric simplicial singular homology Generators: Simplicial maps s :
m ! K , where
m is the standard m-simplex, generate Am .

Boundary operation: Chain group:

@s = sj@
m :

Sm (X ) = Am ; m  0 ; Sm (X ) = 0; m < 0 :

Morphisms: Continuous f : jK j ! jLj on polyhedra give f via f  s and the Simplicial Approximation Theorem.

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165

5.8 Bordism homology Generators: Singular `submanifolds'; that is, continuous maps  : M m ! X , for m-manifolds M m with boundary @M .

Boundary operation: Chain group:

@ = j@M :

Sm (X ) = C 0 (M m ; X ); m  0 ; Sm (X ) = 0; m < 0 : Morphisms: Continuous f : X ! Y give f via f  .

This is more or less homology theory as Poincare [88] originally conceived it. Nowadays it is more conveniently regarded as a spectral theory (cf. section 6.2), as we shall see later (see page 247).

5.9 de Rham cohomology In a number of ways, this is the most important cohomology theory for global differential geometry and for physical eld theory. Indeed, the naturally important objects for both areas are closely related and carry much information on homotopy properties of the manifolds and bundles involved. The theory is supported on smooth manifolds and for smooth maps, but this is not a serious restriction because of the denseness of smooth approximations to continuous maps. Useful reference texts include Bott and Tu [14], Karoubi and Leruste [56], Kobayashi and Nomizu [61]. The section in the Appendix on manifolds and bundles beginning on page 331 gives a brief review of the basic ideas of dierential geometry from the coordinate-free viewpoint, in terms of dierential forms. Appendix E illustrates how a computer algebra package can help with computations involving dierential forms.

Generators: Dierential m-forms ! 2 m(X ) on a smooth manifold X . Coboundary operation: ! = d!, exterior dierentiation. Cochain group: S m (X ) = m(X ); 0  m  dim X ; S m (X ) = 0 otherwise. Morphisms: Smooth f : X ! Y induce f  on exterior products as the dual of the derivative Tf : TX ! TY . De nition 5.9.1 A smoothly triangulated n-manifold is a triple (M; K; h) consisting of a smooth n-manifold M , a geometric simplicial complex K , and a homeomorphism h : jK j ! M such that, for each n-simplex  2 K , hjjj has an extension h to a neighborhood U of jj in Rn with h (U ) a smooth submanifold of M .

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We observe that K is required to be locally nite (cf. page 153), so M will need to be locally compact since h is a homeomorphism. It is known that every smooth manifold admits a smooth triangulation, by the theorem of Munkres [82]. The dierentiable structure on a manifold is independent of its topology, and indeed there may be many choices (or just one!|cf. the table on page 333). However, remarkably, it turns out that for a triple (M; K; h), where h is the triangulation homeomorphism, the oriented simplicial cohomology of K with coecients in R is isomorphic to the cohomology of real dierential forms on M . Theorem 5.9.2 (de Rham cochain complex) Given a smooth n-manifold M there is a cochain complex ( (M ); d) where  (M ) is the graded algebra of real dierential forms on M and d is the exterior derivative, which is of degree +1. We put m (M ) = 0 for m < 0.  Theorem 5.9.3 (De Rham cofunctor) There is a cofunctor of degree 0, fHm(M; d) j m 2 Zg M O

H  ( ; d) : Man /



GrVec : f

/

fHm(N; d) j m 2 Zg

N 

de ned on smooth manifolds by

f

m

H m (M; d) = imkerdmd 1 ;

dm : m (M ) ! m+1 (M ) ;

and on smooth maps f via the linear map f  : m(N ) ! m(M ) : ! 7! !  (Tf; Tf; : : :; Tf ) where Tf : TM ! TN is the derivative map of f .  m It is common to write d for d when the domain is evident. The cohomology arising from H  ( ; d) is called de Rham cohomology theory. Since it takes values in vector spaces over R, we obtain dimensions as cohomology invariants. In particular, if M is a connected smooth manifold, we nd dim H 0 (M; d) = dim R = 1 ; dim H k (M; d) = dimf0g = 0 for k > n ;
because dim k (M ) = nk for 0  k  n and also dim k (M ) = 0 for k > n. We recognize the number dim H k (M; d) as the kth Betti number of M (cf. page 157). These are topological invariants of M and so independent of any particular triangulation, as we have already anticipated in our notation (cf. Nagano's memoir [84]). We shall use de Rham cohomology some in the sequel; it is the most important theory for physical applications because physical elds are representable as dierential forms, which have a natural product that induces the ring structure on de Rham cohomology.

5.10. GEOMETRIC SIMPLICIAL COHOMOLOGY

167

Ex on de Rham cohomology 1. Consider S1 as a real submanifold of C and let v be a unit tangent vector

eld on it. Show that the dual to v is not exact but is a closed 1-form which generates the 1-dimensional de Rham cohomology. 2. Observe that S1 is also a topological group and actually a Lie group because its operations are smooth; how can this be exploited? 3. Consider S3 as a real submanifold consisting of unit quaternions. Again, we have a Lie group, but this time not abelian; investigate its de Rham cohomology.

5.10 Geometric simplicial cohomology Generators: R-linear maps from the vector space of oriented m-simplices over R to R; giving the dual space (Sm (K )) =
m (K ; R). Coboundary operation: m :
m (K ; R) !
m+1 (K ; R) :  7 !   @m+1 ; that is, the adjoint of the boundary operator from simplicial homology.

Cochain group:


m (K ; R) ; m  0;
m (K ; R) = 0 ; m  0: Morphisms: Continuous f : jK j ! jLj on polyhedra give f  via the Simplicial Approximation Theorem.

De nition 5.10.1 For a simplicial complex K , geometric simplicial cohomology with coecients R, (H  (K ; R);  ) is de ned to be the cohomology of the cochain complex f
m(K ; R); m j m 2 Zg : Here,
m (K ; R) is the vector space of R-valued linear maps on the real vector space

generated by all oriented m-simplexes of K subject to the orientation condition m = m0 if m ; m0 are m-simplices of K with the same vertex set but opposite orientations.

Ex Describe cohomology theories dual to some other homology theories. We have taken the dual of geometric simplicial homology (over R) to enable us to de ne an integral of a dierential k-form over an m-dimensional region homeomorphic to a k-simplex. This gives a linear map from the vector space k (M ) to the vector space
m (K ; R) and it is actually an isomorphism.

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

Theorem 5.10.2 (de Rham) Given a smoothly triangulated manifold (M; K; h) of dimension n, then for each k = 0; 1; : : :; n there is an isomorphism

Z deRham k

: H k (M; d)  = H k (K ; R) :

Proof: This follows from the integration of forms over chains

Z

k

: k (M ) ! k (K ; R) : ! 7!

which is linear and satis es the condition

 

Z

k

=

the general form of Stokes's Theorem.

Z k+1

Z

k

!

d ; 

In fact, the exterior product on forms actually determines a product, giving a graded Grassmann algebra (M ). Moreover, the product factors through cohomology and so yields a ring structure on H  (M; d). The following section elaborates a little.

5.11 More on products The formation of the product on forms can easily be visualized because given an m-form ! and a k-form  we obtain an (m + k) form ! ^ . This is actually an antisymmetrized tensor product of multilinear functionals on a real vector space (cf. Section C.3.1). The tensor product here is itself de ned pointwise by taking the product (in R) of the values of the functionals. The latter process exempli es the situation in arbitrary cohomology theories. Suppose that we have a cochain complex K having coecients in a ring R with unity 1 . Take an m-cochain  and an n-cochain ; then their cup product is the (m + n)-cochain de ned on generators by

 [ : Km+n ! R : hv0 ; : : : ; vm+n i 7 ! (hv0 ; : : : ; vm i)
(hvm+1 ; : : : ; vm+n i) where
is the multiplication in R. By inspection, [ is bilinear and associative with unit the 0-cocycle e0 : v0 7! 1.

Ex The coboundary operator satis es ( [ ) =  [ + ( 1)m  [  (cf. section 5.3; also Hocking and Young [46], p. 307, or Vick [114], Chapter 4).

5.11. MORE ON PRODUCTS

169

In consequence, the cup product behaves well with L respect to cohomology classes and we obtain a graded ring of cohomology groups m2ZH m(K ; R). For smooth triangulable manifolds the simplicial cohomology ring over R = R so formed is isomorphic to the de Rham cohomology ring arising from the exterior product of forms, hence it is not commutative but Grassmannian. The integration of forms on a manifold is a special case of the following product operation. The cap product of an m-cochain  and an n-chain c is: the 0-chain if m > n; the (n m)-chain  \ c for n  m, which is constructed by linear extension over generators hv0 ; : : : ; vn i from

 \ hv0 ; : : : ; vn i = (hvn m+1 ; : : : ; vn i)hv0 ; : : : ; vn m i : We can easily express this in terms of a singular n-simplex  and its front and back faces, as used in the construction of the cup product (cf. page 150):

 \  = h; m in m :

Ex on the cap product 1. For appropriate elements, h [ ; ci = h; \ ci. 2. ( 1)m@ ( \ c) = ( \ @c) ( \ c), whence the cap product passes to cohomology and homology as a map H m (X ; R)  Hn (X ; R) ! Hn m (X ; R), for any ordinary theory with coecients a (commutative) ring R. 3. The cap product is natural with respect to induced maps:

f (f   \ c) =  \ f c : The main occurrence of the cap product of interest to us is in Poincare duality.

Theorem 5.11.1 (Poincare duality) If M is a compact, connected, oriented nmanifold with generator z 2 Hn (M ; R)  = R for the ordinary theory H with coecients R, then D : H k (M ; R) ! Hn k (M ; R) : u 7 ! u \ z is an isomorphism for each k.



See Vick [114], Chapters 4 and 5, for a proof and more details. Alternatively, an ambitious reader might try to prove the theorem directly from the Thom isomorphism theorem 7.3.11.

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

Ex on Poincare duality

1. When R is a eld, H k (M ; R)  = Hn k (M ; R). = H n k (M ; R) and Hk (M ; R) 

2. Seek out induced isomorphisms! A (some would say the) general principle of algebraic topology is that the more algebraic structure our theories have, the easier it is to prove that certain maps do not exist. In this sense, cohomology is better than homology because it has a natural ring structure rather than merely a group structure. Certain theories on certain spaces may have even more algebraic structure. We now present one example of this, which will also be used crucially later: namely, certain ordinary cohomologies of H-groups.

Ex Recall the notion of H-group; see Section 2.3. Since the de nition is somewhat long, we shall not repeat it here; but if you have forgotten any part of it (or if you've never seen it before), go look at it now. At present, you may assume that we are using CW-cohomology; later (Theorem 6.3.6), we shall see that what we are about to do is indeed valid for any suitable ordinary theory.

Theorem 5.11.2 (Cohomology of H-group) Let G be an H-group and let H  be an ordinary cohomology theory with coecients R such that H  (G  G)  = H  (G) H  (G), where we do and will suppress the coecients. Then the multiplication  : G  G ! G induces a morphism, called a coproduct,  : H (G) ! H  (G) H  (G) with the following properties: 1.  is a graded R-algebra homomorphism of degree zero for the cup product; 2.  is associative: ( 1) = (1 );

3. the homomorphism " : H (G) ! R : r1 7! r for all r 2 R satis es (" 1)  = (1 ")  = 1A.

Proof: The coproduct is just the composition of  : H (G) ! H (G  G) with the isomorphism H  (G  G)  = H  (G) H  (G). The veri cation of the properties is a straightforward exercise using the properties of H-groups and cohomology, which you should do.  Ex 1. In this setting, the cup product is similarly induced by
 , where
: G ! G  G : g 7! (g; g) is the diagonal inclusion map.

5.11. MORE ON PRODUCTS

171

2. From the Kunneth formula for cohomology (cf. Ex following Theorem 4.4.17), H ( ; Q ) is one such theory. When would H  ( ; Zp) or H  ( ; Z) be such? 3.  also induces a product on the homology H (G), the Pontrjagin product. Thus H  (G) has additional algebraic structure, that of what is called a Hopf algebra; they were created by Hopf [50] almost exactly in this way. Physicists pursue them under the misleading name `quantum groups' which seems to indicate a pious hope; see Kassel [57] for an introduction. (In fairness, however, we note that physicists frequently do not distinguish between a Lie group and its associated Lie algebra.) The standard mathematical reference for them now is [74], from which we adapt the following structure theorem; cf. also Borel [12]. We say that a graded module is of nite type if and only if each homogeneous module (i. e., all the elements of a given degree) is nitely generated.

Theorem 5.11.3 (Hopf algebra structure) Let R be one of Z, Zp, or Q and assume that H is a Hopf algebra of nite type over R. If R = Zp or Q , then H is isomorphic to the tensor product of an exterior algebra with generators of odd

degree and a in nite-dimensional polynomial algebra with generators of even degree. If R = Z and H is torsion-free, then it is isomorphic to an exterior algebra with generators of odd degree. For convenience, we shall count R itself as an exterior algebra with generators of odd degree.

See the cited references for a proof. If one wishes to be de nite about the last convention, one might take the generators to have degree 1 since there are none and dim ? = 1. Let us consider some abstract examples of each type. An exterior algebra over R with one generator is of the form R[x]=(x2 ). If we take the degree of x as 1,
3 3 2 then R[x ] = (x ) has one generator of degree 3. We say that the height of a generator in an exterior algebra is 2. More generally, we may consider polynomial algebras with generators of arbitrary height. For example, R[x] has one generator of in nite height while R[x]=(xn ) has one generator of height n. For this to be a nite-dimensional part of the polynomial algebra in the theorem, we must have a generator of even degree
with a height n > 2. Thus if we take the degree of x as 1, then R[x4 ] = (x4 )23 has one generator of degree 4 and height 23, so is an acceptable candidate.

Ex For concrete examples, H (Sn; R) is an exterior algebra on one generator of degree n and H  (RP n ; Z2) is a polynomial algebra on one generator of degree one and height n + 1.

Theorem 5.11.4 (Hopf) If G is a nite-dimensional H-group, then H (G; Q ) is

isomorphic to the rational cohomology algebra of a nite product of odd-dimensional spheres.

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

In particular, this applies to all Lie groups. Proof: If G is nite-dimensional, then H (G; Q ) is nite-dimensional, hence isomorphic to an exterior algebra by the Hopf algebra structure theorem. It is now a fact of multilinear algebra that this exterior algebra is isomorphic to the tensor product of exterior algebras on one generator of odd degree, and each of these is in turn isomorphic to the cohomology algebra of an odd-dimensional sphere by the immediately preceding Ex. We nish by applying Ex 2 above. 

Ex Write down versions of the theorem for coecients Zp and Z. Corollary 5.11.5 (Hopf) No even-dimensional sphere of positive dimension can be an H-group.  In fact, only S0, S1, S3, and S7 support H-space structures, but this is a much

deeper result; see [52], for example, for further references. To summarize, cohomology is better than homology because it has an additional operation, the cup product. Certain spaces may have even more algebraic structure on their cohomology: H-groups give rise to Hopf algebras, for example. More generally, any cohomology theory can support several other operations in addition to the cup product. These are de ned as certain natural transformations of the component cofunctors, and their study comprises the theory of cohomology operations. A very brief introduction to them, much in the spirit of this book, is in Lomonaco [63]. (You will need to have studied Sections 8.2 and 8.4 a bit, however, before you can read it.) Since they are de ned categorically, they have natural duals: homology cooperations. (A certain sense of humor is evident here.) More thorough treatments of both are in Gray [38] and Switzer [106].

5.12 C ech cohomology theories As we have indicated above, every locally nite cover A = fUg of a paracompact Top space X determines an abstract simplicial complex KA with typical m-simplex

 = f(U0 ; : : : ; Um) j

m \

i=0

Ui 6= ? for some m  0g :

Evidently the (co)homology of KA would tell us more about the cover than about the space X , unless the cover is in some way unique. For the kind of spaces that we usually have to work with, there is no distinguished cover. However, the locally nite open covers of X are partially ordered by re nement, so we can eectively deal with them all together by taking a limit of their inclusion diagram in (co)homology. Complete technical details can be found in Hocking and Young [46], Chapter 8, and Warner [116], Chapter 4. In essence, each locally nite open cover A contributes its own m-cocycles and there is a natural projection induced by a re nement

 5.12. CECH COHOMOLOGY THEORIES

173

inclusion, A  A0 , which preserves the property of being cohomologous among mcocycles because every element of A is contained in some element of A0 . Thus, each locally nite open cover gives a cochain complex and the property of being cohomologous determines equivalence classes over the dierent chains for each dimension m. A representative in each chain for a given class is a local coordinate for the class and addition of classes is de ned by addition of these coordinates in their own chains. The C ech cohomology group at a given dimension is the group of cohomologous classes at that dimension. The important property is that C ech theories and simplicial theory agree on nite polyhedra, but not necessarily on in nite ones; C ech is an example of a continuous cohomology theory, while simplicial is not. We now provide some of the details, as we shall need them later. Let G be an abelian topological group, taking the discrete topology when there is none more usual. For each m-simplex , an m-cochain f is a continuous map jj ! G, where jj = U0 \


\ Um  X for  = (U0 ; : : : ; Um ). Let C m (A; G) denote the abelian group of all m-cochains with respect to the cover A for m  0 and set C m (A; G) = 0 for m < 0. The coboundary operator is now given by

 : C m (A; G) ! C m+1 (A; G) : f 7 !

mX +1 i=0

( 1)i f jU0 \


\U^i \


\Um+1

for m  0 and is the zero map for m < 0, where U^i indicates that Ui is to be omitted from the intersection.

Ex 2 = 0. We denote the cohomology of the resulting cochain complex by H m(A; G).

Ex 1. This cohomology is a covariant functor in G. 2. Show f is a 0-cocycle if and only if f : X ! G is continuous. In particular, if G is discrete then f is constant and H 0 (A; G)  = G. 3. Compare the eect of dierent topologies on G; for example, consider G = S1  = SO(2) with the discrete and the usual topologies for X = S2. Next, we partial order covers by re nement and then take the direct limit to de ne a cohomology theory of X . Let A0 be a locally nite re nement of A. Then there exists a map  : A0 ! A such that V  (V ) for every V 2 A0 . This  induces a map on simplices because

 = (V0 ; : : : ; Vm ) =) () = ((V0 ); : : : ; (Vm )) ;

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CHAPTER 5. EXAMPLES IN HOMOLOGY AND COHOMOLOGY

thus an induced map

# : C  (A; G) ! C  (A0 ; G) : f 7 ! f   ; and, in turn, nally an induced map on cohomology  : H  (A; G) ! H  (A0 ; G). We must now compare the maps induced by two dierent re nements, so let  and  both re ne A0 ! A. For  = (V0 ; : : : ; Vm 1 ) set

~i = ((V0 ); : : : ; (Vi );  (Vi ); : : : ;  (Vm 1 )) and de ne hm : C m (A; G) ! C m 1 (A0 ; G) by hm (f ) =

mX1 i=0

( 1)i f j~i :

Ex h is a cochain homotopy: hm+1 + hm = m m. It follows that  =   on cohomology, so the direct limit with respect to re nements of locally nite coverings is well de ned and exists. We de ne this to be the C ech cohomology of X :  H  (X ; G) = lim ! H (A; G) :

Ex For discrete G, this is an ordinary cohomology theory with coecients G on paracompact Hausdor spaces.

Chapter 6

Sheaf and Spectral Theories What beckoning ghost along the moonlight shade Invites my steps, and points to yonder glade. |Pope, Elegy to the memory of an unfortunate lady.

This chapter describes some generalizations and extensions of homology and cohomology theories. Using sheaves as coecients is the most general way of allowing the coecients to vary from point to point for an ordinary theory. The category of spectra provides a very general way of constructing homology and cohomology theories which need not be ordinary. The method of spectral sequences provides a way of calculating with any homology or cohomology theory which can be very powerful, but may also be very complicated.

6.1 Some sheaf theory Bringing in the sheaves|Old hymn

Covering spaces are surjective local homeomorphisms, hence have discrete bers. Fiber bundles generalize this by allowing more topological structure on the bers, whereas sheaves allow more algebraic structure. Additionally, sheaves allow some algebraic variability from ber to ber. One motivation for introducing ber bundles is to use their sections as generalizations of ordinary maps to a standard space, the model ber. Similarly, sheaves provide a means for collecting local maps to algebraic objects into global sections. Fibrations are (essentially) ber bundles in the homotopy category, allowing the topological isomorphism type of the bers to vary. Sheaves already provide for the algebraic isomorphism type of their bers (traditionally called stalks) to vary. Thus, one may regard ber bundles as a generalization of covering spaces in one direction and sheaves as a double generalization in another. It is convenient to de ne sheaves in terms of the slightly more primitive notion of presheaves. For subsequent 175

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eciency we shall de ne presheaves and sheaves as functors (cf. Johnstone [55]). But rst we follow Tennison [110] and our intuitive preliminaries to de ne a sheaf space, which eventually turns out to be the essential object that characterizes any sheaf.

De nition 6.1.1 A sheaf space or etale space over X is a surjection p:E ! X that is also a local homeomorphism or etale map: for all e 2 E there exists a

neighborhood Ne of e such that the restriction of p to Ne is a homeomorphism onto its image.

Note that this local condition goes the other way to that of a ber bundle with model ber F , which is a surjection p:E ! !B that is also locally a product with F : for all b 2 B there exists a neighborhood Nb of b such that p Nb  = Nb  F :

Ex

1. Given a sheaf space p : E ! ! X , p is an open map and each ber is a discrete subspace of E . 2. Covering spaces and covering projections (see page 62) are examples of etale spaces and maps. 3. Construct contrasting examples of ber bundles, sheaf spaces and brations over a given space. 4. Partial order for a topology Given a topological space X , then its topology T (X ) has a partial order by inclusion; the poset (T (X ); ) is a category with a morphism U ! V if and only if U  V with U; V open in X . We shall see below that a sheaf space E ! !p X is equivalent to a sheaf of germs of local sections of the surjection p; such sections (lifts of the identity map on the base X ) correspond to the physicists' concept of locally de ned (e.g., by coordinate charts) `E -valued elds' over the space X . De nition 6.1.2 Given a space X with topology T (X ) and any (concrete) category C , a presheaf of C -objects on X is a cofunctor (cf. Ex 4 above)

P (U )

U o

O

O

P : T (X ) /



C: V 

/

UV

P (V )

6.1. SOME SHEAF THEORY

177

When C is Set or one of its subcategories (typically Grp or Ab) we call P (U ) the set of sections of P over U and  the restriction map. A morphism of such presheaves P; P 0 is a natural transformation of cofunctors,  : P ! P 0 ; namely, each inclusion U ,! V in T (X ) is sent to a commuting diagram in C : P (U )  (U ) P 0 (U ) U /

o

O

O

O



V 

/

0 UV

UV

P (V )  (V ) P 0 (V ) /

Ex For any space X there is a presheaf of abelian groups given by (cf. Tennison [110], p. 3) P (U ) = 0 for U 6= X; P (X ) = Z ; XX = 1Z; otherwise UV = constant. In many applications to physics and geometry the presheaves will arise from local sections of a surjection; and they are better behaved than this example in that P (U ) is uniquely determined by fP (U ) j  2 Ag for some open cover fU j  2 Ag of U . Such good behavior is rewarded by the following nomenclature. De nition 6.1.3 A presheaf P of C -objects on X is a sheaf if, for all open covers fU j  2 Ag of open U  X and for all f 2 P (U ) j  ;  agree on U \ U ; 8 ; 2 Ag ; then there exists a unique  2 P (U ) such that for all  2 A, UU () =  : A presheaf morphism between sheaves is called a morphism of sheaves. Ex

1. Investigate presheaves of Z2-objects on Z3. 2. Given any object W in C , the constant sheaf WX is given by WX (U ) = W ; UV = 1W :

3. Given a space Y , the sheaf C ( ; Y ) of germs of continuous Y -valued maps on X is de ned by C (V; Y ) = fcontinuous f : V ! Y g ; UV : C (V; Y ) ! C (U; Y ) : f 7 ! fjU ; for U  V , both open in X .

178

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

4. If, in C ( ; Y ), the space Y is a topological group then we get a sheaf of groups. 5. A topology T (X ) itself yields a sheaf O on X by restrictions to open subsets U X : O(U ) = T (X )jU ; UV (A) = A \ U :

p 6. Given a bundle or a bered space E ! ! B , so p is a continuous surjection, then the sheaf of germs of sections of p, denoted (B; E ) or just (E ), is given by (U; E ) = fcontinuous  : U ! E j p   is the inclusion U ,! B g for U open in B . Also, UV () = jU .

De nition 6.1.4 The stalk Px of a presheaf P at x 2 X is the inverse (or left, cf. page 299) limit taken over open sets U containing x: Px = lim P (U ) : x 2U

Elements of Px are called germs of sections of P .

Ex on sheaves 1. Given a space X with topology T (X ) and a (concrete) category C (especially Set or Ab), there are categories: C Presh(X ) = (C T (X ) )op ; the category of cofunctors T (X ) ! C ; C Shv(X ) ,! C Presh(X ) ; the full subcategory of sheaves; and Shfsp(X ) ; the category of sheaf spaces over X . We often write Presh(X ) or Shv(X ), especially when C is Set or Ab. 2. There are adjoint functors and L: : Shfsp(X ) ! Shv(X ) pqqqqq

q qqq

X MMM x

f

MM p0 MMM

(U; E )

E 



E0 

() /

(U; E 0 ) 

6.1. SOME SHEAF THEORY

179

where

() :  7 !   

a

P L : Presh(X ) /

x 2X 

Shfsp(X ) : 

L( )

/

a

P0 

Px



x 2X

Px0

Moreover, is an equivalence of categories with inverse 1 = Lj Shv(X ) and L, called shea cation, is left adjoint to the inclusion functor Shv(X ) ,! Presh(X ) whereas L is naturally equivalent to the identity. See Tennison [110] for a detailed study of the categories Shv(X ) and Presh(X ); note in particular that if C is Ab, then they admit biproducts and they are abelian categories.

Ex on sheaf spaces p 1. If E ! ! X is a sheaf space, then the stalk of ( ; E ) at x 2 X is naturally

bijective to the ber p (x), and is hence a discrete subspace of E . 2. Interpret and justify the statement: a C k -dierential structure on a topological n-manifold is a subsheaf C k ( ; R) of C ( ; R); (cf. Smith [96]). Mainly we shall wish to use presheaves and sheaves of C -objects when C is a category having a zero object (particularly Set and Grp) so that kernels, cokernels and exactness can be de ned as usual (cf. MacLane [65]). Then, for a presheaf (or f sheaf) morphism F ! G we have the following:

 For all H !g F !f G with fg = 0 there is a unique k making commutative g

H

=

k }

ker f





/

{ {{ { {{ {{

F

f /

G

180

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

 For all F !f G !g H with gf = 0 there is a unique c making commutative H

g

o

c

y yy y y yy yy

G o

f

F

coker f with im f = ker(G ! coker f ) in Presh(X ) or Shv(X ), so there is exactness at G in this diagram if im f = ker g. We also note that:  The inclusion functor Shv(X ) ,! Presh(X ) preserves kernels but not cokernels; cf. Tennison [110], p. 44, Example B.  The shea cation functor L : Presh(X ) ! Shv(X ) is exact but the inclusion functor Shv(X ) ,! Presh(X ) is only left exact; cf. Tennison [110], p. 52, Example 6.10.  If H is a subsheaf of a sheaf F , then there is a sheaf G, unique up to isomorphism, and an epimorphism F ! ! G making exact the sequence 0 ! H ,! F ! ! G ! 0: |



|

We shall need to make frequent use of direct limits and some further use of inverse limits, particularly of abelian groups. Recall that a limit of a diagram in any category is an object and associated morphisms having certain universal properties (cf. xA.2.2 for some ampli cation, also Higgins [42], p. 50 et seq., or Dodson [29], p. 20 et seq.).  A direct limit lim! is an example of a right limit or colimit of a diagram and may be viewed as an ultimate target with attendant morphisms.  An inverse limit lim is an example of a left limit or limit of a diagram and may be viewed as an ultimate source with attendant morphisms. The following examples have been collected to help visualize the process. See also Switzer [106], pp. 118{125, and Dugundji [34], Appendices. The easy way to remember which are which is to observe that direct limits go down the arrow stream and unverse limits go up the arrow stream.

Ex on directed systems 1. lim! fA1 ; A2 g = A1 q A2 , with

A1 ,! A1 q A2 - A1

is the direct limit of two objects with no morphisms, in Set. This limit extends to arbitrary disjoint unions, and in Grp it yields the free product group while in Top we choose the largest topology that ensures continuity of the injections.

6.2. GENERALIZATION TO SPECTRAL THEORIES 2. In Set,

181

a lim ! fA1 ! A2 g = A2

with morphism just a. 3. Given a topological space (X; T ), then in Top, the direct limit of the inclusion pairs of open sets lim ! fU ,! V j U; V 2 T g

`

is constructible from U 2T U . 4. Given a presheaf P on (X; T ), then in Set

UV lim f P ( V ) ! P (U ) j U  V 2 T g = P (?); !

with suitable maps; there is a similar result for presheaves of abelian groups. 5. In Set, lim ! fA ,! X j A  X g = X : 6. In Ab,

lim ! fA  G j A is nitely generatedg = G :

S

7. In Top, given subspaces (X )2J of a space X = 2J X with the weak topology on X , then lim ! X = X : 8. If X is a CW-complex with X n its n-skeleton and fX g its set of nite subcomplexes, then n X = lim ! X = lim ! X :

9. Write out some corresponding results for inverse limits of directed systems.

6.2 Generalization to spectral theories There seem to be two paths to generalizations. One idea now is clear enough: instead of using a xed coecient group (or module) for cohomology, we allow the coecients to come from a presheaf or sheaf, eectively using coecient objects that vary from point to point. Choosing a constant sheaf G should recover ordinary (singular) cohomology with coecients in G, and this is indeed the case for locally trivial X , that is for manifolds. Of course, sometimes we do have continuously varying groups naturally arising; for example, on spaces as pointed homotopy classes and on smooth manifolds as holonomy groups. The formalities to prescribe such theories are also easy to state: in any abelian category (e.g., AbShv(X ) or AbPresh(X )) construct a (co-)chain complex of objects and morphisms and de ne its (co)homology in the usual way as quotients of kernels and images. The work

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

182

comes in checking that the appropriate Eilenberg-Steenrod axioms are satis ed. The standard reference here is Spanier [97], Chapt. 6; cf. also Tennison [110], Chapt. 5, Warner [116], Chapt. 4, or Wells [117], Chapt. II. Another approach depends on the notion of a spectrum (cf. G. W. Whitehead [118] for the original constructions) and ts well with the prominence that we give to homotopy theory. For, a theorem of Brown [16] shows that reasonable candidate cofunctors for cohomology are representable as homotopy classes of pointed maps into standard spaces. In consequence, all cohomology theories on spaces homotopy equivalent to CW-complexes are representable by this means, as spectral theories; these include K-theories, stable homotopy and cobordism. We shall follow Switzer [106] (based on Boardman [11] via Adams [1], Chapt. 13) g of spectra in which CW ] is embedded and for the construction of a category Spec to which suspension S admits an extension which is, moreover, invertible there. From the outset we required our theories to be natural with respect to suspension and we deduced from the axioms that S induces isomorphisms in homology. The importance of Spec is that the suspension functor is invertible on it, hence giving a ] . Spectra are Z-indexed sequences in genuine and very useful generalization of CW CW  that have built in suspension simplicity at each index. Morphisms of spectra are sequences of cross linking cellular maps which for all large enough indices yield a commuting ladder, with the proviso that two cross linkings are equivalent if they eventually agree for all large enough indices. This rigamarole is necessary in order to handle properly the cohomology of in nite CW-complexes, of which noteworthy examples are noncompact spacetimes. Precisely we have: De nition 6.2.1 A spectrum E is a sequence n En+1 j n 2 Zg fSEn ,!

in CW  such that each n is a (cellular) homeomorphism onto a subcomplex of (or, an embedding into) En+1 .

Ex on spectra

1. Equivalently, one may de ne a spectrum via maps En ! En+1 . What properties should these maps have? 2. Given a pointed CW-complex X , we de ne the spectrum on X via  S nX for n  0; (E (X ))n =  otherwise. This gives a functor E : CW  ! Spec. 3. Up to homotopy equivalence in CW  , every sequence n En+1 j n 2 Zg fSEn ,! determines a spectrum (Switzer [106], x8.3).

6.2. GENERALIZATION TO SPECTRAL THEORIES

183

A subspectrum F of a spectrum E , so we have Fn  En for all n 2 Z, is called a co nal subspectrum of E if for all cells en  En there exists m such that S m en  Fn+m . Intuitively, all information from En is suspended into Fn+m . The game now is to choose morphisms of spectra such that any spectrum is equivalent to any of its co nal subspectra; then we have the right structure to handle cohomology groups via direct limits. We have begged the question by introducing subspectra, for in a concrete category we want an object F to be a subobject of an object E if the obvious inclusion F ,! E is a morphism. Accordingly we make the following

De nition 6.2.2 A morphism of spectra f : E ! E 0 is an equivalence class []

of commuting cross linking maps

n 0 En j n 2 Zg fFn !

in CW  from co nal subspectra F of E , with respect to the relation (F; )  (G; ) if and only if there is a co nal subspectrum H  F \ G  E and commuting cross n E 0 j n 2 Zg which agree with the restrictions of  and to H . links f Hn ! n

SHn UUUU

SGnA @@ UUU AA U UUUU @@ AAS n UUUU @@ Sn UUUUUU AAA @ UUUU SFn Sn SEn0 o



/

`

n

*

/

Hn+1 UUUU 

n

n

Gn+1@

0n @@ UUU @@ UUUUn+1 @@ @@n+1 UUUU @ @@ UUUU UUUU @@ @ UU Fn+1 En0 +1 n+1 

o



/

_





*

/

Ex On morphisms of spectra 1. Draw, for a xed n 2 Z, the other commuting diagram involved in the de nition of a morphism of spectra.

2. Co nality is preserved under nite intersections and arbitrary unions, and it is transitive. 3. Find a spectrum having the co nal subspectrum fFn = fg j n 2 Zg. 4. Any cellular map  : X ! Y de nes a morphism of spectra via S n  : S n X ! S nY for the spectra on the spaces (de ned above).

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CHAPTER 6. SHEAF AND SPECTRAL THEORIES

5. Every spectrum E is partitioned into sequences, called cells of E , of the form e = fen; Sen; S 2 en ; : : :g ; where en is a cell in En . 6. Given spectra E; E 0 and commuting cross links

n E 0 j n 2 Zg fEn h! n  in CW , then (cf. Switzer [106], x8.13) for all co nal F  E there exist co nal F 0  E 0 such that for all n 2 Z we have hn Fn  Fn0 . The composite morphism of two morphisms of spectra E f! E 0 g! E 00

is denoted by

E gf! E 00 and de ned to be the class gf = [ ] where f = [] and g = [ ]. The category so determined is denoted by Spec.

Ex on composition of spectral morphisms

1. The composition rule is well-de ned and associative. 2. Inclusion (and restriction) cross linking maps induce appropriate morphisms. Theorem 6.2.3 (Spectral co nal inclusions are isic) In Spec, co nal inclusions are isomorphisms. in En and for all cells en  En Proof: Given a co nal inclusion F ,!i E , then Fn ,! there exists m such that S m en  Fn+m . Also, from the identity morphism 1F we obtain  Fn ; jn : En ! Fn : en 7 ! en ifif een  n 6 Fn ;

which de nes a morphism j = [(F; )] = [] with n = jn . Evidently ji = 1F . It remains to show that ij = 1E ; that is, (E; 1E )  (F; i) : This is because F = F \ E is co nal in E , we have (F; 1F ), and ijF = 1F = 1E jF .



Corollary 6.2.4 Any two co nal subspectra of E are equivalent to E in Spec, and hence equivalent to each other.   Now we have Spec containing CW , but we really are interested only in gener] , so we need a de nition of homotopy in Spec alizing the homotopy category CW which restricts to what we already have in CW  . This is achieved by extending smash products to Spec while suitably respecting co nality.

6.2. GENERALIZATION TO SPECTRAL THEORIES

185

Ex on smash products and co nality 1. Given a spectrum E and a pointed CW-complex X , then

S (En ^ X )  = (S1 ^ En ) ^ X  En+1 ^ X ; hence there is a well-de ned spectrum E ^ X .

2. Given co nal F  = E and a pointed CW-complex X , then F ^ X is co nal in E ^ X.

Theorem 6.2.5 (Spectral smash functor) There is a well-de ned functor E ^X

(E; X )

^ : Spec  CW  /



Spec : (f;c)

(E 0 ; X 0) 

f ^c /

E0 ^ X 0 

where f ^ c = [ ^ c] with f = [].



Ex Check the functorial properties and extend this functor to a product on Spec. We now wish to de ne homotopy in Spec. Intuitively, we replace the Cartesian product by the smash product; thinking of spectral poltergeists or Ghostbusters will help to remember this. To prepare, consider I + = [0; 1] q , the unit interval with a disjoint basepoint. The natural inclusions f0g ,! I + - f1g induce morphisms 0 E ^ I + i1 E E i!

in Spec.

De nition 6.2.6 A homotopy in Spec is a morphism h : E ^ I + ! F; and we say that two morphisms

E f! F g E in Spec are homotopic if there is such an h with hi0 = f and hi1 = g.

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

186

Ex on homotopy in Spec 1. Homotopy is an equivalence relation in Spec. 2. The `spectrum on a space' functor E : CW  ! Spec is an embedding:

E (X )

X 



Y 

E () /

E (Y ) 

cf. page 182 for E .

In the literature, it is common to omit the E ( ) when spectralizing diagrams in CW  ] ; however, care is needed when we come to spectral homotopy groups. and CW Homotopy equivalence of morphisms in Spec is preserved under composition. We g the homotopy category induced by homotopy in Spec. In a sense, denote by Spec g  in Spec g because certain maps get suspended invisibly. we partially dissolve CW For example, in the functorial embedding

E : CW  ! Spec ; homotopy inequivalence may disappear under suspension. Nevertheless, most of the desirable features of CW  persist into Spec; in particular, we have available to us now:

   

wedge products, because they commute with suspension in CW  ; cellular construction, by attaching cells fen; Sen; : : :g; homotopy extension property, from E ^ f0g+ to E ^ I + ; suspensions S , by de ning (SE )n = En+1 ; but now with inverse

(S 1 E )n = En 1 ;  homotopy groups; by de ning

n (E )  = lim ! n+k (Ek ) ;

 weak homotopy equivalence, since f : E ! E 0 induces  n (E 0 ); for all n; f : n (E ) =  co brations, such as inclusions of subspectra.

6.2. GENERALIZATION TO SPECTRAL THEORIES

187

Note carefully that [ ; ] in Spec is a direct limit of a directed system of [ ; ] in

CW  .

These notions allow many useful results in CW  to be extended, by formally similar proofs, to Spec. In particular: Theorem 6.2.7 (WHE persists in Spec) In Spec, a morphism f : E ! E 0 is a weak homotopy equivalence if and only if it is a homotopy equivalence.  So, `up to homotopy is good enough' extends to Spec. g we have (i) there is a natural equivalence Corollary 6.2.8 In Spec E ^ S1  = SE ; (ii) each [E; E 0 ] admits an abelian group structure with bilinear composition.

Proof: Switzer [106], p.141{143.  Theorem 6.2.9 (Exact sequences in Spec) Given the following diagram commuting up to homotopy in Spec:

h

G

g /

H



h /

K



E

E0

f /

E 0 [f CE 





/

with ; ; homotopy equivalences, then, for any spectrum F , there are exact sequences  [F; H ] h!  [F; K ] [F; G] g!   [G; F ] g [H; F ] h [K; H ]

Proof: Switzer [106], p. 143{145.



Ex on suspension spectra

1. The co nal subspectra of S n E (S0) are the spectra S n k E (Sk) for k  0. A morphism of spectra S n k E (Sk) ! F is a cellular map Sk ! Fk n together with all of its suspensions. 2. In Spec, given the sequence

E f! E 0 ,!E 0 [f CE with a representative (F; ) for f , then E 0 is of the same homotopy type as the mapping cylinder spectrum of , M = fEn0 [ Fn ^ I + j n 2 Zg :

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

188

3. Given any pointed CW-complex X , then E (SX ) is a co nal subspectrum of SE (X ) and hence this inclusion induces, for any spectrum E , isomorphisms [SE (X ); S n E ]  = [E (SX ); S nE ] ; we already know that [S n E (S0); E ^ S1 ^ X ]  = [S n E (S0); SE ^ X ] : De nition 6.2.10 For any spectrum E we de ne  the reduced E -spectral homology functor Ee = fEen j n 2 Zg with [S n E (S0); E ^ X ]

X

] Een : CW /



Ab : f

(1^f ) /

[S n E (S0); E ^ Y ]

Y





 the reduced E -spectral cohomology cofunctor Ee = fEen j n 2 Zg with [E (X ); S n E ]

X

] Een : CW /

O



Ab : f

E (f ) /

[E (Y ); S n E ]

Y 

Ex One can add coecients to any spectral theory. Given a spectrum E and an abelian group G, de ne a new spectrum EG with EGn = M (G; 1) ^ En 1 and mappings 1 ^ n 1 : M (G; 1) ^ En 1 ^ S1 ! M (G; 1) ^ En . Here, M (G; 1) is a Moore space; see page 261f. We shall see later (page 266) how to recover ordinary homology and cohomology with coecients as spectral theories, and how to construct other famous spectral theories such as K-theory (page 229) and (co)bordism (page 247). For now, we continue with the general theory, heading toward one of its most famous examples. Theorem 6.2.11 (Reduced theories extend to Spec) For a spectrum E , the ] , satisfypair E~ and E~  de ne reduced homology and cohomology theories on CW g ing the wedge axiom. Moreover, these theories extend to Spec by putting [S n E (S0); E ^ F ]

F

g Een : Spec /



Ab : f G 

(1^f ) /

[S n E (S0); E ^ G] 

6.3. SPECTRAL SEQUENCES

189 [F; S n E ]

F

g Een : Spec /

O



Ab : f G 

/

E (f )

[G; S n E ]

Proof: The details are given in Switzer [106] p. 145{149. The de nition and extension of E~ n is forced by the following isomorphism for any spectrum E and pointed CW-complex X : [S n E (S0) ^ X; E ]  = [E (X ); S n E ] : 

Notation There is a natural inclination to write n(E ^ X ) for Een(X ) and n (E ^ F ) for E~n (F ) since both are (spectral) homotopy classes of morphisms

from (spectral) spheres.

De nition 6.2.12 For the particular spectrum E = E (S0)|sometimes abbreviated to S0 or S|the theory E (S0) is denoted S0 or S and called stable homotopy theory S , since for a pointed CW-complex X we have k nS (X ) = n (E (S0) ^ X ) = lim ! n+k (S ^ X ): The dual theory, stable cohomotopy theory S , is the corresponding cohomology theory, with k n k Sn (X ) = lim ! [S ^ X; S ] : Ex on stable theories

1. S is not ordinary: S 1 (S0)  = 4 (S3)  = Z2.

] induce isomorphisms in S . 2. Weak homotopy equivalences in CW ] (hence on T] 3. S and S agree on CW op).

6.3 Spectral sequences In studying homotopy theory we have seen that while in general the groups n (X ) are rather dicult to compute (there is no compact, simply connected, incontractible space for which they are all known), they have the greatly redeeming qualities of behaving very nicely for products and brations. On the other hand, their approximations the homology groups Hn (X ) are much easier to compute, but, even for products, the behavior of the latter is much more complicated (as evinced by the Kunneth formula). For a bration  : E ! ! B , it seems reasonable that Hn (E ) should be some kind of product of Hn (B ) and Hn (F ), where F is the model

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

190

ber; but what kind precisely? We shall now begin our assault on this citadel, albeit a bit indirectly. It will turn out to be convenient to consider a more general setting than CWcomplexes and their skeletons, that of spaces X with a ltration. This is a Zindexed nested sequence of subsets of X , fXp j p 2 Zg, for which we require that

X=

[ p

Xp ; Xp  Xp+1 ;

each Xp is closed in X , and every compact subset of X is contained in some Xp . (This last is a technical requirement to provide for the taking of limits later.) Also,

X 1 = lim Xp =

\ p

Xp = ? :

Let  : E ! ! B be a bration in which B is a CW-complex. De ne Ep =  B p where B p is the p-skeleton of B . Note that Ep is not in general the p-skeleton of E , but that Ep does de ne a ltration of E . This will be our model example. We wish to compute H (X ) for an unreduced homology theory H . We shall proceed by a double approximation scheme. First, we regard Hm (Xp ) as an approximation to Hm (X ). This gets better as p increases. Secondly, we regard Hm (Xp ; Xp r ) as an approximation to Hm (Xp ). This gets better as r increases. We shall investigate each in turn, but rst recall that even if Xp = ? for p < 0, Hm might not be zero for m < 0. In order to compare the various Hm (Xp ), where we regard m as xed and p as variable, we introduce the groups

Fp q = im(Hm (Xp ) ! Hm (X )) ; where p + q = m and the map is induced by the inclusion Xp ,! X . Some sort of double indexing is necessary to keep track of both m and p, and this one is not only elegant but (as we shall see) very useful. Since

X= Thus

[ p

Xp ; Hm (X )  = lim !p Hm (Xp ) : Hm (X ) =

[ p+q=m

Fp q :

Now observe that since the Xp are ordered there is an induced order on the Fp q in which Fp 1 q+1 precedes Fp q and the bottom is F 1 m+1 = 0. Thus we have the series Hm (X ) 


 Fp q  Fp 1 q+1 


 F 1 m+1 = 0 : This is called a ltration of Hm (X ). The ltered degree is p, the complementary degree is q, and the total degree is m. This provides what is usually called a bigraded ltration of H(X ).

6.3. SPECTRAL SEQUENCES

191

Note that (as an algebraic fact) a ltration only determines Hm (X ) up to some extensions. Thus group theory suggests that we should consider the consecutive quotients

Fp q =Fp 1 q+1

as the fundamental objects to study. However, these are de ned in terms of maps into Hm (X ), which we do not have to hand.

Lemma 6.3.1 Consider the triple (X; Xp; Xp 1) and the associated pairs (X; Xp), (Xp ; Xp 1 ), and (X; Xp 1 ). The exact sequences for each of these interlace to produce the commutative diagram with exact row and column Hm+1 (X; XOp ) @

Hm (Xp 1N)

i2

Hm (Xp ) 

/

NN NN NN i1 NNNNN N

OOO OOO DOO OOO OO

Hm (Xp ; Xp 1 ) '

j /

i

Hm (X ) &



where @ and D are connecting homomorphisms from the pair (X; Xp ) and the triple (respectively), the other maps are induced by inclusions, Fp q = im i , and

Fp 1 q+1 = im i1  = im i2 :



Now we compute

Fp q =Fp 1 q+1 = im i = im i1 = (Hm (Xp )= ker i ) = im i2 = (Hm (Xp )= im @ ) = ker j (by exactness) = im j = im j @ (by interchanging the order of quotients) = im j = im D : (6.1) This avoids using maps into Hm (X ), but now we need and lack Hm (Xp ). So we trace this information into the second approximation. We approximate Hm (Xp ) with Hm (Xp ; Xp r ) and thus j with

jr : Hm (Xp ; Xp r ) ! Hm (Xp ; Xp 1 ): Hence we approximate D with Dr : Hm+1 (Xp+r 1 ; Xp ) ! Hm (Xp ; Xp 1 ) :

In the interests of simplicity (and sanity) we shall not multiply-index jr and Dr in the obvious way (or any other). Note that this Dr goes between sequences as

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

192

D does, rather than within one sequence as the other possible candidate does. To help keep track of all this we name the images involved Zpr q = im (jr : Hm (Xp ; Xp r ) ! Hm (Xp ; Xp 1 )) Bpr q = im (Dr : Hm+1 (Xp+r 1 ; Xp ) ! Hm (Xp ; Xp 1 )) and also name their quotient Epr q = Zpr q =Bpr q : Let us check now and see how the approximation to Fp q =Fp 1 q+1 is doing. Lemma 6.3.2 We have the following inclusions: r+1 r 1 0 = Bp1 q 


 Bpr q  Bpr+1 q 


 Zp q  Zp q 


 Zp q = Hm (Xp ; Xp 1 ) :

It may be amusing to note the formal resemblance to nested intervals here. Proof: Clearly, 0 = Bp1 q and Zp1 q = Hm(Xp; Xp 1). To see that Bpr q  Bpr+1 q , we use the de nition to construct Hm+1 (Xp+r ; XVp ) O

VVVVDr+1 VVVV VVV

Hm (Xp ; Xp 1 ) h

hhh hhhh hhhh Dr

*

4

Hm+1 (Xp+r 1 ; Xp )

r and commutativity means that we have the desired inclusion. Similarly, Zpr+1 q  Zp q is equivalent to commutativity of

Hm (Xp ; Xp rU) UU O

UUUjUr UUUU U

Hm (Xp ; Xp 1 ) *

i iiii iijir+1 i i i  i 4

Hm (Xp ; Xp r 1 ) and we leave the veri cation that Bpr q  Zpr q as an exercise. Thus we may de ne

Zp1q = lim Zpr q =

\ r



Zpr q

r = [ Br Bp1q = lim B p ! q r pq r Ep1q = lim ! Ep q :

Of course, we must identify these more precisely if we wish to compute with them.

6.3. SPECTRAL SEQUENCES

193

Lemma 6.3.3 In the notation introduced in Lemma 6.3.1, using equation (6.1), we have Zp1q = im j and Bp1q = im D. Moreover, Ep1q = Zp1q =Bp1q :

Proof: In outline, using the de nitions, Zpr q = im (jr : Hm (Xp ; Xp r ) ! Hm (Xp ; Xp 1 ))

converges (as r ! 1) to

Zp1q = im (j : Hm (Xp ; ?) ! Hm (Xp ; Xp 1 ))

and converges to

Bpr q = im (Dr : Hm+1 (Xp+r 1 ; Xp ) ! Hm (Xp ; Xp 1 )) Bp1q = im (D : Hm+1 (X; Xp ) ! Hm (Xp ; Xp 1 )) :

We leave the last statement as an exercise.



It is not true in absolute generality that homology commutes with inverse limits. However, the theories that we consider and the cases in which we shall apply the preceding will always obey commutativity, usually because the ltration will be bounded below in the sense that Xp = ? for all p suciently small.

Corollary 6.3.4 Fp q =Fp

= Ep1q . 1 q+1 



It is now clear that the Epr q are the fundamental objects of our double ap-

proximation scheme. For a xed r, there is some additional algebraic structure that is crucial in computations. To obtain it, we consider a situation similar to that in Lemma 6.3.1, but now using the exact sequences associated to the triples (Xp ; Xp 1 ; Xp r ) and (Xp ; Xp r ; Xp r 1). Somewhat intuitively one may regard these respectively as `the rth approximation to the 1st approximation' and `the 1st approximation to the rth approximation'. Using the exact sequences for the associated triples (Xp ; Xp 1 ; Xp r 1) and (Xp 1 ; Xp r ; Xp r 1) as well, we may interlace these exact sequences to obtain

Hm (Xp ; Xp rR 1 ) R Hm (Xp 1 ; XpR r ) R

RRR RRRjr+1  RRR RRR RRR

Hm (Xp ; Xp r ) 

/

RRR RRR R Dr RRRRRR RR

Dr+1

Hm 1 (Xp r ; Xp r 1) )



Hm (Xp ; Xp 1 ) (

jr  /

194

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

where, e.g., Dr+1 is the connecting map of the triple (Xp ; Xp r ; Xp r 1 ), and the unmarked arrows are induced by inclusions. Essentially the same computation as for equation (6.1) shows that

Zpr q =Zpr+1 = Bpr+1r q+r 1 =Bpr r q+r 1 : q 

Combining this isomorphism with the de nitions and inclusion, we may create

Zpr q =Bpr q ! ! Zpr q =Zpr+1 = Bpr+1r q+r 1 =Bpr r q+r q 

,! Zpr rq+r 1=Bpr r q+r 1 :

1

This composition de nes a map

dr : Epr q ! Epr r q+r with so that

1

r ker dr = Zpr+1 q =Bp q r im dr = Bpr+1 q =Bp q

im dr  ker dr and (dr )2 = 0 ; r ; dr ) is a bigraded chain whence dr is a dierential, like a chain map, and (E complex (i.e., the 2-dimensional version of a chain complex). Amazingly, we can now compute to nd: r+1 r
r
ker dr = im dr = Zpr+1 q =Bp q = Bp q =Bp q r+1 r+1 = Zpr+1 q =Bp q = Ep q :

r+1 ; dr+1 ) is the homology of (E r ; dr )! This complete sequence fE r ; dr g That is, (E  pq of bigraded chain complexes in which the homology (as above) of each is the next, constitutes the algebraic monster known as a spectral sequence. The term `spectral' is because of the homology relation, an historical allusion. In general, they do occur in other contexts, both abstract algebraic and topological; ours will all t the preceding construction. We refer to McCleary [69] for a survey (with details) of the most important spectral sequences in algebraic topology, and a guide to others. 1 or E 2 ) and would like to have Frequently, we know an early term (usually E  1 E . In this case we try to work out enough dierentials to gain intermediate terms r . On the other hand, sometimes we know E 1 and try to work back to get an E  early term. In order to keep track of what's going on overall, we shall view the terms as living in 3-dimensional (p; q; r)-space and consider slices of constant r; then we can display progress on pictures of the resulting (p; q)-planes.

Ex Observe that d2 is a `knight's move', and dr is a generalized one. Satisfy

yourself that the sketch in Figure 6.1 represents the situation. Of course, the d3 and d4 shown there really live in the r = 3 and r = 4 planes, respectively; we drew them together purely for illustrative purposes.

6.3. SPECTRAL SEQUENCES

195

q 6

q q q q} q q q q Z Z d q q q qY q q HHqZZq qY qk q qHHqZZq q HHq Q Z q qHHq QqQdQq qY H q Zq H q q q q qQQq qHHq 4

3

-

p

Figure 6.1: Picturing spectral sequences Note that arrows for d2 lie along lines of slope 12 , and in general those for dr along slope 1 r r . Since p + q = m, we may say that H (X ) is along limit lines of slope 1. Our objective was to compute H (X ), and this has been achieved up to some group extensions. Nevertheless, we say that our spectral sequence converges to H (X ), and write Epr q =) H (X ) ; re ecting the double approximation scheme we have employed by the double arrow. As an example of what can be done with spectral sequences, we have the following. Theorem 6.3.5 (Atiyah-Hirzebruch) Let H denote an ordinary homology theory and let h denote any homology theory satisfying the wedge axiom, De nition 4.5.1. If X is a CW-complex, then there exists a spectral sequence Epr q =) h (X ) with Ep2 q  = Hp (X ; hq ()) :

Proof: We use the p-skeleton X p as our ltration of X . This ltration S is bounded below since X p = ? for p < 0 and converges to X since X = p X p. We know from previous work that X p is obtained from X p 1_

X p =X p =



1

by attaching p-cells, so that

Sp

where  indexes the p cells. Thus, applying our construction above, Ep1 q = hp+q (X p ; X p 1 )  = hp+q (X p =X p 1; ) = hp+q (_ Sp; ) 

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

196

= M hp+q (Sp; )  M 

=



hq (S0; )

= Cp (X ) hq () where Cp (X ) is the group of cellular p-chains which occurred in computing the homology of CW-complexes. From our construction of the general spectral sequence before, it is easy to see that

d1 = D1 : hp+q (X p ; X p 1) ! hp+q 1 (X p 1 ; X p 2 ); is the connecting homomorphism of the triple (X p ; X p 1 ; X p 2).

Following this through the isomorphisms at the beginning of this proof, we nd that d1 maps to

@ 1 : Cp (X ) hq () ! Cp 1 (X ) hq (): Hence, as desired, Ep2 q  = Hp (X ; hq ()).



1 above depends on a choice of CW structure for X , E 2 does Note that, although E  not. Thus, in theory, all one needs to know in order to compute any homology theory on CW-complexes is ordinary homology and the coecients of the exotic theory. r and In practice, one needs a bit of luck rst to obtain the intermediate terms E r their dierentials d for r > 2, and then to solve the extension problems where necessary to pass from Fp q to h (X ). Computer algebra software is sometimes able to compensate for bad luck. For our next illustration of the power of spectral sequences over ordinary theories, we deduce their universal topological invariance:

Theorem 6.3.6 (All ordinary theories agree) Finite simplicial homology (and thus cohomology) is a topological invariant. In fact, all ordinary theories agree on CW-complexes.

Proof: Take h to be nite simplicial homology. Then in the Atiyah-Hirzebruch spectral sequence,



; Z) for q = 0; = Hp (X ; hq ())  = Hp (X pq  0 for q = 6 0:

E2

We see that there is only one nonvanishing row in the r = 2 plane, so all dierentials dr for r  2 are zero homomorphisms and the spectral sequence collapses:

Ep2 0  = Fp 0 =Fp = Ep10 

But

Fp

= Ep 1 1 1 1 =Fp 2 2  1

11 :

=0

6.3. SPECTRAL SEQUENCES which implies that

197

Ep10  = Fp 0 :

Moreover, it follows that By de nition of Fp q we obtain

Fp q = 0 for q 6= 0:

Ep2 0  = hp (X ) for nite simplicial complexes X considered as CW-complexes jX j. It is now clear that all ordinary homologies agree on X , or on jX j, so in particular nite simplicial homology is a topological invariant.



Next, we indicate the proof of the Hurewicz theorem as promised earlier (page 141).

Proof (Hurewicz isomorphism): Recall that stable homotopy is the reduced homology theory

k n+k k nS (X ) = lim ! n+k (S X ) = lim ! [S ; (S X )] ;

so we have maps

n (X ) ,! nS (X ) : Hence we deduce that for simply connected X , for any xed n, k (X ) = 0 for k < n () kS (X ) = 0 for k < n : By suspension, for (n 1)-connected X , k (X )  (6.2) = kS (X ) for 1  k < 2n 1 and there exists : 2n 1 (X ) ! 2Sn 1 (X ). Now consider the spectral sequence

(Atiyah-Hirzebruch) Note that

Ep2 q  = Hp (X ; qS ()) ! S (X ) : Ep2 0  = Hp (X ; Z) ; S

im  (X ) Ep10  = Fp 0 =Fp 1 1 = im Sp(X p ) p p 1




! pS (X )
! pS (X )

pS (X )


; im pS (Xp 1 ) ! pS (X ) and we have an inclusion Ep10 ,! Ep2 0 . Whence the Hurewicz map =

pS (X ) ! Hp (X ; Z)

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

198 factors through Ep10 and


Ep10  = im pS (X ) ! Hp (X ; Z) :

Applying the suspension result (6.2) above, yields the Hurewicz theorem (and we do 1 ! H1 directly by an elementary argument). 

Theorem 6.3.7 (Only two nonzero rows in Ep2 q ) If Ep2 q has only two nonzero rows, Ep20 = Ep2 n , then clearly

Ep1q = 0 if q 6= 0; n ;

and dn+1 is the only non-zero dierential. Moreover, n+2 = E n+3 =


= E 1 : E  

Proof: In E2 we have nontriviality only at q = 0; n so 2 2 2 2 0 d! Ep2 0 d! 0 and 0 d! Ep2 n d! 0 ;

Ep3 0 = Ep2 0 =f0g = Ep2 0 and Ep3 n = Ep2 n =f0g = Ep2 n ;

etc. The only nontrivial dierentials arise at level n + 1, namely: n+1

0 ! Epn0+1 d ! Epn+1n

10

n+1 ; and ! 0 so Epn+2 0 = ker d

n+1

0 ! Epn++1n+1 0 d ! Epnn+1 ! 0 so Epnn+2 = coker dn+1 here. Above level n + 1 we have

Ep q = 0 unless q = 0 or n; and with zero dierentials; so the homology lters up and is preserved to the limit:

n+2 = E n+3 =


= E 1 . E  



A typical situation is when we have a spectral sequence with

Ep2 q = Hp (X ; Rq ) 1. where the coecient group Rq is a qth homology group, and we are interested in E 2 Filling in the known information, we may have an Ep q diagram with relatively few non-zero entries to work from. Suppose that the non-zero entries are as indicated r will have at least the same zero entries as in Figure 6.2. Evidently, for r > 2, E 2 E . Hence, our rst possible nonzero dierentials are

d3 : E33 q ! E03 q+2 for q even

6.3. SPECTRAL SEQUENCES

" q

.. . 8 7 6 5 4 3 2 1 0

199

.

.

.

F8

A8

B8

F6

A6

F4

A4

F2

A2

F0

A0

6.. 6 6 6.. 6 6.. 6 6 6

0

1

2

3

6 -

B4

-

4

5

6

7

p!

-

8




Figure 6.2: Example of a sparse Ep2 q diagram which appear as homomorphisms

d3 : Aq ! Fq+2 for q even. 1 has only nitely many zero entries (this happens if our Suppose further that E spectral sequence is known to converge to the homology of a nite dimensional manifold). Then for suciently large m we must have for all q  m,

E34 q = E04 q+2 = 0 : Hence for large enough m,

d3 Am = Fm+2 : With speci c information about the Aq and Fq+2 it may be possible to identify the action of the d3 homomorphisms on generators. Clearly, E34 q = 0 if d3 : Aq ! Fq+2 is monic. The next, and only other, possibly non-zero dierentials are

d5 : Bq ! Fq+4 for q = 4; 8; : : :; and again for suciently large m we must have d5 Bm = Fm+4 :

200 Then everything gels:

CHAPTER 6. SHEAF AND SPECTRAL THEORIES Ep1q  = Ep6 q :

Before pursuing further applications, we pause to dualize the constructions and discuss brie y cohomology spectral sequences. We shall hit the highlights and leave most of the details for exercises or reference by the reader; see e. g., Spanier [97] and Switzer [106]. We write a cohomology spectral sequence as fErp q ; dr g and observe that now we want dr : Erp q ! Erp+r q r+1 but preserving Er+1 still as the homology of Er . In Figure 6.1 we reverse the p q , Z p q , etc., are now de ned directions of the arrows. Since such objects as E1 1 as inverse limits , our cohomology theories must be well behaved with respect to inverse limits in order that analogous relations still hold. The ordinary, C ech, and de Rham theories are examples of well-behaved theories, as are most of those we consider; exceptions will be explicitly pointed out. Both types of spectral sequence are well behaved with respect to products, and we shall consider fErp q ; dr g as0 0a bigraded ring with derivations dr whenever convenient. If a 2 Erp q and b 2 Erp q , then ab 2 Erp+p0 q+q0 and dr (ab) = (dr a)b + ( 1)p+q a dr b : In the cohomology version of the Atiyah-Hirzebruch spectral sequence, we have maps Erp q ! h (X ), both additively and multiplicatively, and E2p q  = Erp q =) h (X ) as rings. The additional work needed for a proof is mostly a careful but tedious checking of details, and we refer to e. g. Hilton [43] or Switzer [106].

Ex on the con guration space of identical particles Selig [93] used spectral sequences to compute the cohomology of the con guration space of three identical particles moving in certain manifolds M . So we seek H  (C3 (M ); Z) where Ce3 (M ) = M  M  M n fall diagonalsg and C3 (M ) = Ce3 (M )=S3; so Ce3 (M ) ! ! C3 (M ) is a regular cover. Here S3 is

the symmetric group of permutations of three objects. In this case, the spectral sequence (cf. MacLane [65], p. 342) has   E2p q = H p S3 ; H q (C~3 (M ); Z) and

Ep q =) H p+q (C3 (M ); Z) : For example, when M = R3 , the E2p q term has the nonzero groups and generators shown in Figure 6.3, where deg a = deg c = 2 and deg b = 4. Explicitly, we have the following generators for n = 0; 1; 2; : : ::

6.3. SPECTRAL SEQUENCES

4 3 " q 2 1 0

4 3 " q 2 1 0

201

6 6 6 6 6 6 6 6 6

Z3

Z3


-

-


-



Z2

Z2

Z2

Z2

6

-

Z6
-

0 1 2 3 4 5 6 7 8


p!

Z

Z2

Z6

Z2

6 6 6 6 6 6 6 6 6

f

c

0

ac a

1 1

bf

2

bc b

3

4

p!

abc

6


-

-


-

-

ab 5

6

7

b2
-

8




Figure 6.3: Part of the E2 -plane; groups (top) and generators (bottom)

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

202

   

abn for E24n+2 0 , bn for E24n 0 , bnc for E24n+1 2 , and abnc for E24n+3 2 .

Now, C3 (R3 ) is nite dimensional so, for large enough n, H n (C3 (R3 ; Z)) = 0 and the rst possibly non-zero dierential is d3 , operating from row q = 2 to row q = 0. Hence, for large enough n, we get exactness in row q = 2 so

d3 : (E34n+1 2  = Z) ! E34n+4 0 : bn c 7 ! 3bn+1 ; d3 : (E34n+3 2  = Z2) ! E34n+6 0 : abn c 7 ! abn+1 :

(6.3) (6.4)

It is also the case that d is a derivation (cf. Selig [93])

d3 xy = (d3 x)y + ( 1)deg xd3 y : Therefore, for large enough n, we obtain from (6.3)

d3 (bn c) = (d3 bn )c + ( 1)4n bn d3 c = 0 + bn d3 c = 3bn+1 ; and from (6.4)

d3 (abn c) = (d3 abn )c + ( 1)4n+2 (abn )d3 c = 0 + abn d3 c = abn+1 : Hence d3 c = 3b, which destroys the q = 2 row in E4 since the d3 -chain passing through E3p 2 is exact. Moreover, d3 , being monic, has a Z2 image in groups of row 0, nicely reducing it. Hence the E4 term is reduced to that shown in Figure 6.4, where E44n+3 4 is generated by bnf . Repeating the dimensionality argument for d5 , the only other possibly non-zero dierential, yields

E54n+3 4  = Z3 ! E54n+8 0 ; d5 (bn f ) = (d5 bn )f + ( 1)4n bn d5 f = 0 + bn d5 f = 3bn+2 : Therefore d5 f = 3b2, which destroys the q = 4 row (by injectivity of d5 ) and the q = 0 row for p  8 (by surjectivity of d5 ). Hence the only surviving entries in E6 are

" 0 q

6 6 6 6 6 6 6-

Z 0

Z2

1

2

p!

Z3

3

4

-

6.3. SPECTRAL SEQUENCES

4 " 3 q 2 1 0

203

6 6 6 6 6 6 6 6 6

Z3

Z3

6


-

-


-

-

Z3
-

0 1 2 3 4 5 6 7 8


p!

Z

Z3

Z2

Figure 6.4: Part of the E4 -plane

M

nn R3 R4 S3 S4

0 1

Z Z Z Z

2

3

4

5

6

7

Z2 Z3 Z2 Z Z3 Z2 Z Z3 Z2 Z2 Z12 Z2 Z

Table 6.1: Cohomology for con guration spaces of 3 identical particles

204

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

  so E1 = E6 and we conclude that the only cohomology is H 0 (C3 (R3 ); Z)  = Z; H 2 (C3 (R3 ); Z)  = Z2 ; 4 3 H (C3 (R ); Z)  = Z3 : Similar analyses are given by Selig [93] for H  (C3 (M ); Z) when M = R4 , S3, and S4; he found the results summarized in Table 6.1. He also classi ed the principal bundles over these spaces. More generally, one may consider the con guration space Cn (M ) = Cen (M )=Sn of n identical particles in M , where one takes the n-fold product of M by itself, removes all the diagonals, and mods out by the symmetric group on n objects. When M is a compact, orientable surface, the fundamental group of Cn (M ) is called the nth braid group of M [9]. It appears in knot theory, amongst other places, and has recently attracted new attention in both mathematics and theoretical physics.

6.3.1 Review and moral tale Let's look back over the construction of spectral sequences. First of all, a spectral sequence dr E r E = fEpr q ! p r q+r 1 j r 2 N ; p; q 2 Zg is fundamentally an algebraic entity living in the space Z3; the (p; q)-plane being viewed as horizontal. At xed r, the horizontal sequence through any point (p; q; r) 2 Z3 given for n  0 by

s : n 7! (p nr; q + nr n; r) singles out a chain complex from E . That is, the homomorphisms de ned by dr between the modules at the point of s compose pair by pair to give the zero

homomorphism. Brie y we write

(dr )2 = 0 : This is sucient to ensure that at each point of s the departure of the associated chain complex from exactness there is correctly measured by the homology H (Epr q ; d) = ker dr = im dr : The speciality of spectral sequences is that any such lack of exactness gets kicked up to the next higher level, r r Epr+1 q = H (Ep q ; d ) ; in somewhat similar fashion to the way that bureaucracies promote individuals who don't t at their present level. We are interested in Ep1q , up in the penthouse. r in each plane of constant r as chain Horizontal sequences like s allow us to view E complexes proceeding from lower right to upper left with `slope'

6.3. SPECTRAL SEQUENCES

205

(change in q) / (change in p) = (1 r)=r. Hence the handy little diagrams like Figure 6.1 so often drawn. Now, just as promoted bureaucrats may not know what to do when they get upstairs, so also do we have a problem in that, unlike its domain of operation, dr+1 is not necessarily determined in its action by (Epr q ; dr ). On the other hand, our hierarchy is rmly based on democratic principles dictated by the ground oor functions d1 , so we can represent the superstructure as a tower of submodules in the following manner.

Ep2 q = ker d1 = im d1 is a subquotient of Ep1 q .

Ep3 q = ker d2 = im d2 is a subquotient of ker d1 = im d1 and hence ker d2 = im d2  = = K 2 =I 2 with K 2= im d1  2 2 2 1  ker d and I = im d = im d . Next, Ep4 q = ker d3 = im d3

is a subquotient of ker d2 = im d2 and hence ker d3 = im d3  = = K 3 =I 3 with K 3 =I 2  3 3 3 2  ker d and I =I = im d , and so on, referring back at each stage to a subquotient of Ep1 q with ker dr  = K r =I r 1 at (p; q); im dr  = I r =I r 1 at (p r; q + r 1): For variety, we are using K for kernels and I for images instead of Z and B , respectively. Hence we obtain a decomposition: 0  im d1 = I 1  I 2  I 3 





 K 3  K 2  K 1 = ker d1  Ep1 q with

Epr+1 = K r =I r for r = 1; 2; : : : : q 

Pushing our bureaucratic analogy to its logical conclusion we view the submodules in these terms:

 K r consists of the survivors of promotion to level r (r-Kreeps ?);  I r consists of those bounded by stage r (they have achieved their level of incompetence and are to be factored out: the r-Incompetents);  K 1 = \1r=1 K r , consists of immortal promotees;  I 1 = [1r=1 I r consists of all the eventual bounders.

Observe that K 0 = Ep1 q is not included in K 1 . It follows that I 1  K 1 and so we obtain the limiting quotient (those with an afterlife role?)

Ep1q  = K 1 =I 1

CHAPTER 6. SHEAF AND SPECTRAL THEORIES

206

which is really what we have been after all along, the earlier terms being merely approximations. Note that any tower of (bigraded) submodules 0 = I 0  I 1  I 2 





 K 2  K 1  K 0 = Ep1 q with isomorphisms at each (p; q) given by

1 r = Kpr q 1=Kpr q  = Ipr r;q+r 1 =Ipr r;q +r 1 ; hence of bidegree ( r; r 1), determines a spectral sequence fEpr q ; dr g for r  1

with

Epr q = K r 1=I r

1

and r induces dr with the same bidegree:

K r 1 =I r

1

dr /

K r 1=I r

1

O

K r 1=K r 

=

r /

I r =I r

1

where the vertical arrows are well-de ned maps of quotients since I r 1  K r . Some particularly simple cases are worth looking at.

Theorem 6.3.8 (Immediate convergence case) If Ep2 q = 0 when p or q is odd, then dr = 0 for all r  2 and it follows that Ep1q  = Ep2 q : Proof: By de nition,

d2 : Ep2 q ! Ep2 2 q+1 so d2 = 0, since if p and q are even then q + 1 is odd. For r > 2, dr : Epr q ! Epr r q+r 1 so dr = 0, since for p and q even then either p r is odd or q + r 1 is odd. Now, for all p; q, d2 = 0 so we deduce Ep3 q = ker d2 = im d2 = Ep2 q ; hence inductively, for r > 2, Epr q = Ep2 q : So here the tower has, for all r  1, I r = 0 and K r  = Epr+1 = Ep2 q : = ker dr  q  T K r = E 2 . It follows that Ep1q  = 1 pq  r=1

6.3. SPECTRAL SEQUENCES

207

The proposition transcribes into the following bureaucratic result, rst exploited by the CIA.

Corollary 6.3.9 (CIA theorem) If odd people t straight away, then there is no communication after the rst level and most of the superstructure is irrelevant.  This subsection rst appeared as [28], which was written when we rst thought we had nearly nished this book.

Chapter 7

Bundle Theory The Whore on the Snowcrust: In Defense of Bundling |US ballad c 1786

This chapter is not really a leisurely introduction to bundles, but new tricks for the old dog. We shall assume that the reader has already encountered bundles elsewhere (for example, principal bundles in gauge theories in physics), and direct you to Appendix C and the works cited there for more background material. Nonetheless, it is possible to learn bundles ab ovo here, provided that you are willing to do a certain amount of extra work to familiarize yourself with the traditional view as you go along. Given a base space and a group, bundles occur in various forms; but the fundamental idea of what it is all about is captured by the following.

Ex There are just two Z2-bundles (double covers) over S1. The rst is the trivial

bundle

ptrivial : S1  Z2 ! ! S1 : (x; i) 7 ! x

with total space consisting of two copies of S1. The second is the twisted bundle with total space consisting of one copy of S1 folded back on itself once, like a bandsaw blade on a streetcar or a rubberband on a poster. These are shown in Figure 7.1, each picture there being a view from the same perspective. The existence of the nontrivial bundle is due to the nontrivial homotopy type of the base space S1, nicely displayed even by the minimally nontrivial ber Z2|a thing worth remembering. In contrast, there is only one double cover (Z2-bundle) over R1 . (Draw the pictures; play with a bandsaw blade or a rubber band!) We shall return to the analysis of the Z2 bundle over S1 below (page 211), by means of C ech cohomology. We shall see that principal G-bundles over X are given by H 1 (X ; G) and that in this approach global, local, and computational formulae all appear at once. This 209

CHAPTER 7. BUNDLE THEORY

210

Trivial bundle

Twisted bundle

Figure 7.1: Principal Z2-bundles over S1 will be rather dierent from the usual approach, but very soon it will all fall out right in your lap. As the notation just used suggests, we must consider cohomology with nonabelian coecients (tangent bundles are GL(n)-bundles). While the theory for H 2 and higher `groups' is dicult, and not even possessed of a consensus at present, H 1 is relatively easy. The more theoretically minded reader might keep in mind the following question, to which we have no completely satisfactory answer: how about homology; in particular, why not H1 ?

7.1 Elemental theory It is most convenient to approach from the viewpoint of C ech theory. Here and throughout, X will be our xed but arbitrary base space and G a topological group. To cater for nonabelian groups, we shall use multiplicative notation.  ech 1-cocycle on U Let U = fUi g be an open covering of X . Recall that a C with values in G is a collection of continuous local functions fij : Ui \ Uj ! G such that fij = fji 1 in G, fij = fik fkj in G. The second is historically referred to as `the cocycle condition'. Two cocycles f and g are said to be cohomologous if and only if there exists a collection of continuous

7.1. ELEMENTAL THEORY

211

local functions hi : Ui ! G such that

fij = hi 1gij hj :

As usual, H 1 (U ; G) denotes the set of cocycles modulo the equivalence relation (veri cation?) of being cohomologous. Note that although H 1 (U ; G) has a distinguished element (the class of the constant cocycle 1), it is not in general a group.

Ex Give an example to show why not. What precisely is the problem in general? A covering U is said to be simple if and only if each Ui \ Uj is homologically trivial; that is, H~ n (Ui \ Uj ) = 0 for all i; j; n (using ordinary reduced homology with integer coecients).

Ex 1. Read through the examples in Appendix C. 2. Every manifold has simple atlases. 3. Can you characterize spaces which have simple covers? Partially order all coverings by inclusion (each covering precedes any re nement of it), and then de ne H 1 (X ; G) = lim H 1 (U ; G) : !

What makes these eciently computable is the following extraordinary theorem. Theorem 7.1.1 (Leray) If U is simple, then H 1(U ; G) = H 1(X ; G). 

This theorem is usually presented for abelian G, in which case it's actually valid for the entire H  . Proofs of this version may be found in Spanier [97] or Vaisman [112].

Ex 1. For those willing to consult such a reference, check that one need only assume that each component of Ui \ Uj is simply connected. We also leave to the reader (any, now) the extension to the case of sheaf coecients; that is, the de nition of H 1 (X ; S ) where S is a sheaf of (possibly nonabelian) groups. 2. Let X = S1 and G = Z2  = f1g, writing multiplicatively. Choose U1 = S1nfS poleg and U2 = S1nfN poleg. Note that we must use the discrete topology on Z2. Thus there are only four possible cocycles. The two that assign either +1 or 1 to both components of U1 \ U2 are cohomologous, and the two that assign +1 to one component and 1 to the other are also cohomologous. Thus there are two cohomology classes in H 1 (S1; Z2) the rst `trivial' and the second `twisted'.

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The reader may already have recognized the `cocycle condition' as the compatibility condition of the transition functions for a G-bundle. In order to see bundles in the usual way, we must globalize. Recall that we may identify functions and their graphs. In particular, given a 1-cocycle f , each fij can be identi ed with the subset of Ui \ Uj  G which is its graph. (Recall that these cocycles can be de ned only with respect to some covering.) In order to consider cohomologous cocycles we must also study graphs in each Ui  G. Also ` note that, given a covering U of X we can reconstruct X from the disjoint union Ui by means of the inclusions Ui ,! X and the identi cations

Ui \ Uj = Uj \ Ui : Thus, in order to interpret the cocycles ` and the cohomologous relation in terms of graphs, we form the disjoint union (Ui  G) and mod out by the equivalence (verify!) relation determined as follows: rst slot|the relation given above which yields X ; second slot|the relation determined by the cocycle. In other words, if x 2 Ui \ Uj , then (x; g1 )  (x; g2 ) () g1 = fij (x)g2 :

Let us denote the set so obtained by P . Observe that as a set P  = X  G and that there is a natural surjection P ! ! X such that  fxg = G as sets for each x 2 X . As a space , however, P is only `X -locally' homeomorphic to X  G, that is with this topology  Ui  = Ui  G for each Ui 2 U . The map  is clearly continuous ` on P . Moreover, there is a continuous action of G on (Ui  G) which is ( xedpoint) free. It is easy to verify that P inherits a continuous ` free action of G. We shall make the convention that G acts on the right of (Ui  G) by right translation, and thus that P is a right G-space in which the action of G on each set  fxg corresponds to the action of G on itself by right translation. By now it should be obvious (assuming you've already seen them) that P is a principal G-bundle over X . We call P the total space and X the base space. Any representative cocycle for P is called a set of transition functions for P . If G is discrete, P is also called a G-torsor over X ; see page 217 for the most important example.

Ex Using your favorite usual de nition of principal G-bundle (or the one in Appendix C if you wish), complete the veri cation of this claim. For example, verify that each ber  fxg is homeomorphic to G. We now investigate the cohomologous relation. Let f and g be two U -cocycles and assume that they are cohomologous via some family fhi : Ui ! Gg. Denote the two principal G-bundles constructed from f and g by P and Q respectively. De ne

F : P ! Q : p 7 ! phi ((p)) where (p) 2 Ui . Here, as is customary, we have suppressed the ber identi cations with G.

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Ex 1. F is well-de ned (use the cohomologous relation). 2. F is a homeomorphism which preserves bers (use the facts that G acts by homeomorphisms and that each hi is continuous). Summing up, we have achieved a G-bundle classi cation:

Theorem 7.1.2 (G-bundle classi cation) Isomorphism classes of principal G-

bundles over X correspond bijectively with elements of H 1 (X ; G), the trivial class corresponding to the distinguished element. 

Henceforth we shall regard principal G-bundles interchangeably as geometric objects or as (representatives of) cohomology classes, whichever is more convenient.

Ex (The frame bundle, consisting of linear frames over a smooth manifold; cf. p. 351. Historically, this was the primary example.) Let X be a smooth manifold and U = f(Ui ; i )g any simple atlas. De ne a cocycle l by lij (x) = D(i  j 1 )j (x) ; that is, the matrix of partial derivatives of the change of coordinates. Using the chain rule, we see that l is in fact a cocycle. We call l the Leibniz cocycle and its cohomology class the Leibniz class. Clearly the corresponding principal GL(n)bundle is the frame bundle: the lij are its classical transition functions. This example shows that indeed it all falls out at once. Notice all the classical computations and checks that we do not need to make; listing these is a good exercise in `unpacking'.

Ex 1. Look up an approximation theorem somewhere to verify that smooth maps are dense in continuous maps. This means that we can take all maps to be smooth when X and G are both smooth. 2. A principal bundle P over X is trivial if and only if it admits a section; i. e., a map X ! P which commutes with projection. (Cf. Theorem 7.1.6.) Now suppose that H is another (Top) group and that we are given a (continuous) homomorphism f : H ! G. For each X this induces a natural map H 1 (X ; H ) ! H 1 (X ; G) via composition of H-cocycles with the homomorphism.

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Ex Verify that this does preserve the cohomologous relation, as claimed. If P is a principal H-bundle which maps to a principal G-bundle Q, we say that P is obtained from Q by a reduction (think of H  G) or lifting (think of H covering G) of the structure group, or that Q is obtained from P by an extension or prolongation of the structure group. From the de nitions we immediately obtain the next result.

Theorem 7.1.3 (Structure group reduction) For a subgroup H of G, a principal G-bundle admits a reduction to H if and only if there exists a bundle atlas for which the transition functions are H-valued. Proof: Translating from classical into cohomology language, a `bundle atlas with 

H-valued transition functions' is a `G-cocycle which is also an H-cocycle'.

Later we shall give a bundle-theoretic criterion for reduction to a subgroup (page 220). The other problems require the machinery of Obstruction Theory, so they will have to wait until after Chapter 8. For now, we have another example which is suggestive.

Ex on the orientation bundle Consider the map GL(n) ! O(1) = f1g given

by the formula

g : g 7 ! j det det gj It induces a map from GL(n)-bundles to O(1)-bundles, now representable cohomologically as

or : H 1 (X ; GL(n)) ! H 1 (X ; O(1))  = H 1 (X ; Z2) : The image under this map of a principal GL(n)-bundle P is called the orientation bundle or(P ). The bundle P is said to be orientable or nonorientable according as or(P ) is trivial or twisted, respectively. The classical case of course is when P is the frame bundle of a manifold; we shall then write or(X ) = orX for this orientation

bundle. In this case, verify that our de nition matches your favorite (the classical one is obviously easiest).

This brings us to the connection with representation theory, the preceding example being one of the most celebrated instances. In general, any homomorphism G ! H is called a representation of G in H . When H is the general linear group of some (possibly in nite-dimensional) vector space, the representation is said to be linear. We shall be most interested in representations G ! Aut(X ) in the group of self-homeomorphisms of a space X (self-dieomorphisms when X is smooth) which are continuous (smooth). We say then that X is a G-space. See Michor [70] for an appropriate notion of smoothness. Our immediate objective is to obtain a categorical (that is, systematic, consistent, etc.) method of changing the bers in a principal bundle into any (reasonable)

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215

space, while preserving all the essential information of the principal bundle. Somewhat more intuitively, we want to think of a principal bundle as some kind of holder into which we can insert various bers which may be changed at will. If we think of the bers as `vertical' and the base space as `horizontal', then what we wish to do is to make a vertical change while preserving the horizontal information. Comparing with our construction of the topological space P , which was a `horizontal' quotient, we see that what we should do is to take a `vertical' quotient of P  F for some proposed ber F . In order to make parity consistent, we shall always consider only left G-spaces F and write gf to indicate that g 2 G is applied to f 2 F by means of the representation G ! Aut(F ) which makes F a G-space. Clearly, what we must do is to collapse each ber of P to a point. Thus, de ne a right action of G on P  F by (p; f )g = (pg; g 1 f ) and then de ne P [F ] = (P  F )=G, the orbit space of this action. We denote the equivalence class or orbit of (p; f ) by [p; f ] and de ne a projection  : P [F ] ! ! X : [p; f ] 7 ! (p) ; where the second  : P ! ! X comes from the principal bundle as before. We call P [F ] an associated bundle of P . Associated bundles with ber Rn are called vector bundles.

Ex 1. Work out the details for the plain band and the Mobius band as associated bundles of the two principal Z2-bundles over S1. Use R or B 1 for the ber F according as you want open or closed bands, respectively. Figure 7.2 shows the two total spaces for the choice B 1 ; the `middle' circle corresponds to 0 2 B 1 . 2. Verify that this yields a ber bundle with ber F and group G according to your favorite de nition. 3. If you didn't do so before, write all this out in terms of local functions and coverings (bundle charts). 4. If a ber bundle E and a principal bundle P have the same transition functions, then E  = P [F ] for some F . 5. If R denotes the category of (left) G-spaces and BX the category of G-bundles over X , then each principal G-bundle P over X de nes a functor R ! BX . 6. If P denotes the category of principal G-bundles over X and B[F ] that of G-bundles with ber F , then P ! B[F ] : P 7! P [F ] is also a functor. 7. Can you characterize H 1 (X ; G) or even H 1 ( ; G) categorically? A good solution should provide higher cohomology objects and classifying spaces (viz. infra) for them.

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216

Trivial bundle

Twisted bundle

Figure 7.2: Total spaces of associated bundles over S1 with ber B 1 8. Observe that using linear representations of G provides us with vector bundles, as you may easily verify using (as usual) your favorite de nition. Let n T X denote the nth exterior power of the cotangent bundle of a smooth manifold X , and recall that orX denotes the orientation bundle obtained from the frame bundle of X . When O(1) has its natural representation on R, then nT  X  = orX [R]. Let F : R ! R be a functor from the category of left G-spaces to itself. If you can write down explicitly what F does, then you can write down transition functions for the result of `applying F to a G-bundle'. This will also be true if F maps some subcategory of R into itself; for example, G-modules|vector spaces with a G-action.

Ui \ Uj fij G /

rep

/

Aut(F ) F /

Aut(FF )

Here rep above the arrow indicates the action of G on F via the representation and F above the arrow indicates the action of F on morphisms in R. By suppressing explicit mention of the representation, we obtain the very useful recipe for F(fij ): apply the functor to the transition maps of the bundle P [F ] to get the transition maps of the bundle P [FF ]. Classically, these are called `change-of-coordinates' formulas.

Ex 1. Observe that the orientation bundle was obtained from the frame bundle by applying det =j det j to the transition functions. The concept of orientation

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217

can similarly be extended to any associated bundle of any principal GL(n)bundle. 2. Open any dierential geometry book and nd a bundle (newer books) or a change-of-coordinates formula (older books); the more complicated, the better. Write down the change of coordinates formula for the bundle and/or state the bundle for the formula. (Several repetitions provide an excellent review of multilinear algebra! Hint: start with the frame bundle.) We shall see some more examples of functors in the next section. Another application of our bundle theory is in describing what classically were called `local systems of coecients' or `twisted coecients' for ordinary (co)homology. We consider a G-bundle E = P [F ] over X .

Ex

1. If F is discrete, then the structure group can be reduced to 0 (G)  = G=G0 , with G0 denoting the component of the identity. (Hint: for any orbit G=H we have G0  H since all G-orbits in F are of dim 0.) In this case P [F ] is a covering space of X . 2. When F has algebraic structure, P [F ] is a sheaf over X . Which sheaves are not obtained this way? Using the natural projection G ! ! 0 (G), we may induce a map H 1 (X ; G) ! H 1 (X ; 0 (G)) : P 7 ! SP called the Steenrod map; SP is called the Steenrod class of P . Note that (any representative of) SP is also a principal 0 (G)-bundle or 0 (G)-torsor (see page 212) over X . We now de ne the bundles of coecients associated to E to be S P [n (F )] for each n. These are sheaves of groups over X which were classically denoted by fn (F )g or by fn (Ex )g where Ex = p fxg. These SP [n (F )] are indeed the classical `local' or `twisted' coecients. De nition 7.1.4 Given any cohomology theory h, we say that E is h-orientable if and only if SP [h (F )] is trivial as a bundle (or sheaf). Note carefully that this does not imply that SP is a trivial principal bundle in general.

Ex

1. SP is trivial if and only if E is H  -orientable for ordinary cohomology with coecients Z. 2. SP is trivial if and only if SP = 0 2 H 1 (X ; 0 (G)). Remark 7.1.5 Just in case it was not obvious, we state explicitly that the action of 0 (G) on n (F ) is induced by the action of G on F and note again that 0 (G)  = G=G0 is an abelian group for all G. Similarly, we can de ne orientability with respect to a generalized homology theory h .

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Ex Z.

S P = 0 if and only if E is H -orientable for ordinary homology with coecients

Frequently, one does not wish to emphasize the principal bundle and ber as

G-spaces. Then we usually say something like `E is a ber bundle over B ' or `E is a G-bundle over X ' or whatever the base space is. Some notation is frequently used. If E is a ber bundle over X , then a map s : X ! E with s = 1X is called a

section. We denote the set of all (continuous, or smooth, in that category) sections by (E ), or (X; E ) if necessary.

Ex Vector bundles always have sections, but (general) ber bundles need have

none.

If E is a G-bundle, then there is a natural representation of G on (E ) given by sg(x) = (s(x))g. Since E = P [F ] for some P and F , one may regard the representation of G on (E ) as being decomposed by means of P into that of G on F . This can be generalized even more by gluing together bundles over various spaces which are themselves strata of some larger strati ed space, as in Davis [24]. The generic situation here is for F to be nite dimensional and (E ) to be in nite dimensional, so this is potentially of great importance in analyzing these most intractable representations. It is not known which representations can be so decomposed. In addition, there are large numbers of secondary problems with regard to the eciency and beauty of any particular such decomposition, the minimum necessary class of bers and base spaces, etc. We should also note here the vital use of bundles in quantum eld theories. All bundles are considered over one base space called the spacetime. The bundles themselves are the quantum elds and come in two types: a principal bundle is an interaction eld, and certain of its associated vector bundles are the matter elds for that interaction. Sections of the vector bundles are wavefunctions for those types of matter. The interaction potential is a connection on the principal bundle, whose curvature is the intensity of the interaction. Associated with the connection on the principal bundle is an induced covariant derivative on each vector bundle, which gives the coupling of that matter eld and the interaction eld. The structure group is called the gauge group and expresses the local symmetry of the interaction. Elements of the individual bers are called phases or internal states , and entire bers are points of the elds. Each of the four known interactions (gravitational, electromagnetic, weak, and strong) has been expressed in this form, and some of them have been uni ed via prolongation of structure groups. Paraphrasing t'Hooft [109], while we don't have the master yet, at least all the keys are cut from the same blank. Finally, we hope to have convinced you that this section presents a better way of conceptualizing bundles and manipulating them globally. Not only is nothing lost that was there in the classical formula-based approach using transition functions,

7.1. ELEMENTAL THEORY

219

computations using them are clearly outlined and enormously streamlined by our new framework. Happy bundling!

7.1.1 Pullbacks Because we have de ned bundles via a cohomology theory, there are two kinds of `pullback' operation immediately available: one induced from maps between base spaces and the other induced from `coecient homomorphisms'. We shall consider these in order. Let f : Y ! X . From general cohomology theory there is an induced map f  : H 1 (X ; G) ! H 1 (Y ; G) for every (Top) group G. Recall that at the cochain level this is given merely by composition where  is a G-cochain on X .

 7!   f ;

Ex f  as above is a natural transformation of cofunctors. For associated bundles, we thus de ne

f (P [F ]) = f P [F ] where P is a principal G-bundle over X and F is a (left) G-space sometimes called the model ber. This bundle is called the pullback bundle of P [F ] along f . Certain pullback bundles have special names: if A  X and  : A ,! X , then

P [ F ] j A =  P [ F ] is called the restriction of P [F ] to A. In the special case of a point x 2 X the restriction is called the ber over x and is frequently denoted by P [F ]x.

Ex

1. If f : Y ! X and if E ! ! X is a bundle, then (f  E )Y = Ef (Y ) . This is frequently regarded as an identi cation, with (we hope) due respect for the usual dangers.

2. Explain why there cannot be any general method of pushing bundles forward. (Hint: recall the de nition of a function.) For the second kind of `pullback', let F 0 ! F be an equivariant map of (left) G-spaces. Any principal G-bundle P is a functor, so there is an induced map P [F 0 ] ! P [F ] .

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Ex This map commutes with projections to the base space. When F 0  F and the equivariant map is the inclusion, P [F 0 ] is said to be a subbundle of P [F ].

Ex For each x 2 X there exists an equivariant embedding P [F 0]x ,! P [F ]x. Thus, each ber of a subbundle may be regarded as a subset of the host bundle's ber at the same point. The special case of subgroups and principal bundles is most interesting. It provides the bundle-theoretic and topological criteria promised earlier for structure group reduction (and another solution to Ex 2, p. 213).

Theorem 7.1.6 (Structure group reduction) Let H be a closed subgroup of G

and P a principal G-bundle. P has a reduction to H if and only if P [G=H ] has a section.

Proof: If P has a reduction to H , then there is an H-cocycle of transition functions for P . But this forces P [G=H ] to be trivial, whence there exists a section (in fact, many if G 6= H ). Conversely, if P [G=H ] has a section, then the section de nes a subbundle Q of P with model ber H . The action of G on P restricts to an action of H on Q which is free and by right translation. Hence Q is a principal H-bundle, and the inclusion Q ,! P means that the class of Q in H 1 (X ; H ) maps to the class of P in H 1 (X ; G) under the cohomology mapping induced by the inclusion H ,! G, as desired.  Corollary 7.1.7 If n (G=H ) = 0 for every n < dim X , then every principal Gbundle over X has a reduction to H .

This will be proved later; see page 281. We state it here because it provides a very quick and easy proof of the following theorem, which we hope will thus help motivate the reader to study Chapter 8.

Theorem 7.1.8 (Riemannian metrics) Every manifold has a Riemannian metric.

Proof: By using the polar decomposition, GL(n)=O(n) is contractible. (Verify this!) Then a choice of reduction to O(n) as the group for the tangent (or cotangent) bundle is equivalent to choosing a Riemannian metric.  You might wish to compare this with the usual proof using partitions of unity. Note carefully that reducibility of the structure group is not equivalent to reducibility of the representation on the model ber.

7.1. ELEMENTAL THEORY

221

Ex The representation of G on F is irreducible if and only if there are no invariant subbundles of P [F ]. Similarly to what we did for C ech cohomology with abelian coecients in Chapter 5, one may show that H 1 ( ; G) is a homotopy invariant. It now follows from category theory [41, x30] that this functor is representable. This means that there exists a space BG and a principal G-bundle EG ! ! BG such that we have a natural isomorphism H 1 (X ; G)  = [X; BG] of pointed sets, for all X having the homotopy type of a CW-complex. BG is called the classifying space of G. What BG looks like, we have no idea now; however, as certain games enthusiasts would surely agree, it is indeed a really Big Fiber Gun to wield if we can but get our hands on it! Before trying to construct such a monster, let us consider further whether or not it is worth the eort.

Ex Compare with the earlier notion of classifying space in Section 2.2. Suppose we have a given principal G-bundle P over X . Then there is some map f : X ! BG such that P corresponds to [f ] 2 [X; BG]. Since this correspondence is a natural isomorphism, surjectivity implies that P  = f EG. Thus, classifying

spaces allow us to express every bundle as a pullback of a universal bundle by a classifying map. This in turn means that we can separate the properties of being a G-bundle from those of being a G-bundle over X . In other words, we can use bundles to compute information about X ; or more precisely, about its homotopy type.

Ex Stop and think; this is very important! Thus, an explicit construction of BG (and EG) will, in theory, allow us to calculate explicitly all universal properties of G-bundles. Surely this is enough motivation to produce a very concrete construction! So, what do we know a priori about BG and/or EG? Well, we must have BG = EG=G so we need make only EG. Now EG must be a free G-space, so each orbit must be homeomorphic to G. Thus we can make EG by gluing copies of G together appropriately. How? To get some information, consider the homotopy groups of EG. Recalling that n (EG) = [Sn; EG], let f : Sn ! EG be a map and de ne f~ : Sn  G ! EG : (x; g) 7! f (x)g; that is, we have composed the map (x; g) 7! (f (x); g) : Sn  G ! EG  G, with the action of G on EG. Regarding Sn  G as the trivial G-bundle

CHAPTER 7. BUNDLE THEORY

222 over Sn, we have the commutative diagram

Sn  G pr1

Sn 

f~

/

ww f www w ww ww ;

pf

/

EG p

BG 

Observe that, by construction, (pf ) EG = Sn G. Since BG is the classifying space we must have pf  , the trivial map Sn ! BG. If p were a bration and EG were a CW-complex, then we could conclude that f   and thus that EG ' . (Recall the Whitehead theorem.) In any case, this computation is certainly suggestive; `abstract nonsense' also encourages the idea of a BG as a `homotopy quotient of a point' (cf. Section 30 in Herrlich and Strecker [41] on representable functors). Two more pieces of intuition are useful here. One is that since EG is to be `universal', it should be obtained by gluing together orbits (copies of G) in `all possible ways'. The other requires that we recall the notion of join. Given two spaces X and Y , the join X  Y is intuitively obtained by connecting each element of X to each element of Y with a copy of [0; 1]; cf. p. 26f.

Ex Give a precise formulation before proceeding.

7.1.2 The Milnor construction We shall be interested in multiple joins of a space with itself, so we'll only de ne this case formally. It turns out to be most convenient later to use barycentric coordinates. Thus we de ne the n-fold join of a space X with itself to be the set of all 2(n + 1)-tuples (g0 ; t0 ; g1; t1 ; :::::; gn ; tn ) 2

P

n a

i=0

X I

such that ni=0 ti = 1, modulo the following equivalence relation: (g; t)  (g0 ; t0 ) if and only if ti = t0i for each i and gi = gi0 for each i with ti > 0, so only `endpoints' are identi ed. Note carefully that if ti = t0i = 0, then we may have gi 6= gi0 even though (g; t)  (g0 ; t0 ). Thus certain copies of X are collapsed to points. We must now provide this set with a topology in order to have a space. Observe that we have two families of `coordinate' functions: ti maps into [0; 1] and gi maps ti (0; 1] into X . Note this latter is necessitated by the indeterminacy when ti = 0. We de ne the topology to be the weakest (compactly generated) which makes all these functions continuous.

Ex The n-fold join of any space with itself is (n 1)-connected.

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223

Thus, if we de ne EG(n) to be the n-fold join of G with itself and then de ne EG = lim ! EG(n), then we shall have a homotopy point which is an excellent candidate.

Ex De ne a (right) G-action on EG coordinate-wise on the gi only, and by de ning BG = EG=G obtain a principal G-bundle.

It turns out that universal bundles are characterized by having a contractible total space. Thus this construction, due to Milnor [71], provides us with the classifying space desired. Remark 7.1.9 A good, clean proof of this characterization is not available; it is considered to be `folklore' by the experts.

Ex 1. If G is a CW-complex, then so are EG and BG. 2. Let G = Z2 and develop the following table. In it we have used  to denote the singleton space and the trivial group, is S1, is the crosscap and 1 is the in nite crosscap.

E Z2(0)

jj S0  B Z2(0) jj RP 0

E Z2(1)

jj S1

B Z2(1) jj RP 1

E Z2(2)






N B Z2(2)












RP 2






jj S2 jj

E Z2(n)






E Z2

Nn

N1

B Z2(n)












RP n






RP 1

jj Sn jj

jj S1

B Z2

jj

3. E Z ' R and B Z ' S1. 4. E S1 ' S1 and B S1 ' C P 1 . 5. Apply the homotopy exact sequence of a bration to EG ! ! BG. When is BG a K (; n)? (See De nition 8.2.1 for K (; n).) 6. Can you determine B S3 ' BSU (2)? 7. Recall the (in nite) Stiefel varieties and Grassmannians from De nition 3.9.1, page 100. Show that Vk (F n ) is a principal O(k)-, resp. U (k)-, resp. Sp(k)bundle over Gk (F n ), and that BGL(k)  = BO(k)  = Gk (R1 ) ;

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224

BGL(k; C )  = Gk (C 1 ) ; = BU (k) 

and similarly for H . [Hint: page 101, Ex 9.] 8. Can you determine B Zn for each n? g . What does this say about the functors 9. [SX; BG]  = [X; G] for every X in CW

and B here? 10. Is there any reason why the Milnor construction cannot immediately be extended to H-groups (De nition 2.3.1) by just doing everything only up to homotopy?

Theorem 7.1.10 (Bundle classifying functor) B : Grp ! Tg op is a functor

which preserves products.

For the proof, the product part is easy (it uses only the universal properties), and you can nd the induced morphism part (should you need it) on p. 204 of Switzer [106]. It is a theorem of Iwasawa [53] that every connected Lie group admits a deformation retraction onto a compact subgroup. This means that we need consider only classifying spaces BG for compact G in order to handle all Lie groups; in particular, BGL(n) ' BO(n) and BGL(n; C ) ' BU (n). This must be taken with a grain of salt, as may be seen in the case of the Lorentz group [Ex!]; see page 249.

Ex (for category theory fans) Express the Milnor construction in (categorical) simplicial language. Use this to extend the construction to more general G; for example, small categories. How far can you go?

Problem Observe that BG plays the same role for nonabelian cohomology that a K (; 1) does for abelian cohomology (cf. p. 274, Ex 1). Extend this to nd objects corresponding to K (; n) for n > 1. Collect a Fields Medal afterwards (if under 40).

We have seen (if you did the exercise) that the homotopy groups of BG are just those of G shifted. Unfortunately, the homology and cohomology are much more complicated and we defer consideration of them. (The solution has been known to make theorists start gibbering upon hearing the word `calculation'.) A summary of examples needed is in Section 7.4.

7.2 Stabilization This section concerns the concept of stabilization. The basic idea is that if you have enough dimensions to play with, then you can avoid pathologies which are not inherent in the problem. We have already encountered some manifestations of this

7.2. STABILIZATION

225

concept in our study of (generalized or exotic) cohomology theories; for example, stable homotopy groups. Here we wish to examine stability in the context of ber bundles. First some preliminary constructions. If E is any bundle with ber F , then we saw earlier that E = P [F ] for some principal bundle P . We begin by considering the problem of nding such a P for a given E . One might think that this problem has a unique solution, at least in some sense. Let P be a principal G-bundle over X and let F be any space. Assume that G acts on F by the trivial representation (that is, every element of G acts as the identity map of F ). Then P [F ]  = X  F as bundles; but, unless P is trivial itself, P [F ] is technically not a trivial G-bundle. In this case it is clear that the principal bundle which we should associate to X  F is X  1, where 1 denotes the trivial group of one element. This motivates us to try and nd an intrinsic de nition of the principal bundle which should be associated to a ber bundle E over a space X . Now, part of being a ber bundle is that we must be given a principal G-bundle P over X (that is, some G-valued transition functions), as well as a model ber F of course. As was just seen, however, G may be excessively large. Thus we want to nd a reduction of P to a principal H-bundle Q where H is `as small as possible' among subgroups of G. This is clearly a universal-type problem, and as usual category theory assures us that a solution exists and is `essentially' unique.

Ex By considering particular G-valued U -cocycles for coverings U , show that `essentially' means `up to conjugation'. Thus the isomorphism type of H as an abstract group is uniquely determined. (Hint: If P is given by a U -cocycle g and  is the given representation of G on F , start by looking at the subgroup of Aut(F ) generated by g. Then consider other equivalent covers U 0 . ) We shall call the principal H-bundle Q constructed above the (minimal) associated principal bundle of E .

Ex 1. Let X be a smooth manifold and let P be the principal GL(n)-bundle determined by its Leibniz class: the frame bundle (cf. page 351). Regard R as a GL(n)-space via the representation det : GL(n) ! GL(1) : Using your favorite de nition of orientability, show that X is orientable if and only if the associated principal bundle of P [R] is X  1. 2. One may use a similar method to discuss orientability of vector bundles in general.

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226

3. Let X be a smooth manifold with tangent bundle TX . Recall that we may regard TX as classi ed by a map X ! BO(n) via the frame bundle. Show that X is orientable if and only if there exists a lifting

BSO(n)
0. Applying Leray-Serre, we have

E2pq  = H p (B ; H q (Sm; R)) =

 H p(B; R) if q = 0; m; 0

otherwise.

Thus there are only two possibly nonzero rows in the r-plane

q

6

m

-

0

-

p

and only dm+1 6= 0. Thus the spectral sequence collapses and we have  : Em+2  =


 = E1

Ex 1. Deduce that we obtain a long exact sequence, the Gysin sequence, 




! H n (E ; R) ! H n m (B ; R) dm!+1 H n+1 (B ; R) p! H n+1 (E ; R) !


in which dm+1 (x) = e [ x for a certain e 2 H m+1 (B ; R) called the Euler class of the bration. For a generator u 2 H m(Sm; R) = E20 m, e = dm+1(u). 2. The Gysin sequence applied to the Hopf bration S1 ,! S2n+1 ! ! C P n yields  R[e]=(en+1 ) H  (C P n ; R) = where e 2 H 2 (C P n ; R) is a generator. It follows that

H  (C P 1 ; R)  = R [ e] :

3. How about the other Hopf brations? 4. If the Gysin sequence were valid for n = 0, we could apply it to S0 ,! Sn ! ! RP n . Carry this out formally; can you justify it? 5. To obtain the Gysin sequence, it suces that the ber be a homology sphere.

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244

7.3.2 The Wang sequence

 If the bration is F ,! E! !p Sn with n > 0 and h is any cohomology theory, then the E2 -diagram has only two possibly nonzero columns (p and q have interchanged roles) and the only nonzero dierential is dn : En0 p ! Enn p n+1 . Thus the spectral sequence collapses again.

Ex

1. We wind up with a long exact sequence, the Wang sequence 

n hm n+1 (F ) ! hm+1 (E ) !


:


! hm (E ) ! hm(F ) d! 2. Compute the ring H  ( Sn; Z). (Hint: separate the cases for n even and n

odd.) 3. To obtain the Wang sequence, it suces that the base be a cohomology sphere.

7.3.3 Transgression and the Serre sequence

 p Suppose that in the bration F ,! E! ! B we have Hp (B ; Z) = 0 for 0 < p < n, Hq (F ; ) = 0 for 0 < q < m, and that F and B are connected. Then the E 2 -diagram becomes

q

6

pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp pp p p p p p p p p p-p

6

m



n

-

p

and in a certain range, the only nonzero dierentials go all the way from the p-axis to the q-axis, dp : Epp 0 ! E0p p 1 .

Ex

1. We obtain a nite exact sequence, the Serre sequence    H Hn+m 1 (E ; ) p! n+m 1 (B ; S [ ]) ! Hn+m 2 (F ;  ) !  Hn+m 2 (E ; ) !


! H2 (B ; S[]) ! H1 (F ; ) !  H (B ; S [ ]) ; H1 (E ; ) p! 1 p where  = d is called the homology transgression and S is the Steenrod class.

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245

2. Write down the cohomology version and obtain the cohomology transgression.  = dq+1 : H q (F ; ) ! H q+1 (B ; S[]) where S is the Steenrod class. 3. More generally, in the cohomology version q ! E q+1 0 ;  = dq+1 : Eq0+1 q+1

and similarly for homology. 4. Combine Gysin and Wang and compare to Serre. Anytime an inverse to transgression exists, it is called (confusingly) suspension. There is some relation between the two suspensions, if that helps any.

Ex 1. Find any such relation, even in special cases. [See p. 363 of Switzer [106] for an example.] 2. Find such a relation which can be used to prove the Freudenthal suspension theorem.

7.3.4 The Leray-Hirsch theorem

In this theorem, H denotes a ring spectrum, like ordinary cohomology theory (but with arbitrary ring coecients, not just Z).

Theorem 7.3.10 (Leray-Hirsch) Let (F; F0 ) ,! (E; E0 ) !p B be a bration with B connected. If e1 ; : : : ; er 2 H (E; E0 ) are such that  e1 ; : : : ;  er 2 H  (F; F0 ) form a free basis for this H  ()-module, then H  (E; E0 ) is a free H  (B )-module with basis fei g under the action b
 = p (b) [  : Proof: First observe that the bration is H -oriented via the basis in each ber. Now de ne a cohomology theory

h (X; A) = H  (X; A) H  () H  (F; F0 ) which will satisfy exactness since H  (F; F0 ) is a free H  ()-module. Via linearity,

de ne

X;Y : h (X; Y ) ! H  (E jX ; E jY [ E0 jX ) : x  ei 7! p x [ j  ei where Y  X  B and j : (E jX ; E0 jX ) ,! (E; E0 ). Observe that  is natural with respect to inclusions. Assuming, as we may, that B is a CW-complex,  induces a

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246

map from the skeletal spectral sequence of B and h to the (Leray-Serre) spectral sequence of the bration. Denoting these by Er and Er , respectively, we have morphisms pq pq pq pq : F pq ! F pq r : Er ! Er ; p+r q r+1 dr , and these diagrams commute: such that dr pq r = r

Erpq+1

pq r+1 /

Erpq+1 O

O

=

=

H (Erpq ) (pqr ) H (Erpq ) /

F p+1 q

0 /

pq E1

F pq

1 /

/

pq 1

pq

p+1 q 1

0 /

pq E1 0 F pq F p+1 q 1 Moreover, recalling the de nitions of  and , this commutes:

0







/

/

E1pq

/

pq 1

/

/

E1pq O

O

=

=

[H  (B p ; B p 1 ) H  () H  (F; F0 )]p+q Bp ;Bp 

H p+q (Ep ; Ep 1 [ (E0 )p ) 

1 /

 1



[C p (B ; H  ()) H  () H  (F; F0 )]p+q 

C p (B ; H q (F; F0 )) 

= /

pq Hence pq 1 , and thus all r , are isomorphisms. Inducting on the rst two diagrams, all pq are isomorphisms. Therefore

B : H  (B ) H  () H  (F; F0 )  = H  (E; E0 )

and fei g is a free basis as desired.

Ex Write down the dual homology result.

7.3.5 Thom isomorphism theorem As usual, B is connected; H is again any ring spectrum.



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247

Theorem 7.3.11 (Thom isomorphism) If (B n ; Sn 1) ,! (E; E0 ) !p B is an oriented ball bundle, then there exists u 2 H n (E; E0 ) such that  u 2 H m(B n ; Sn 1) is a generator over H  (). We call u the Thom class. Moreover, for all q 2 Z there are isomorphisms:  : H q (B ) ! H q+n (E; E0 ) : b 7 ! p b [ u ;

 : Hq+n (E; E0 ) ! Hq (B ) : x 7 ! p (u \ x) :

Proof: Apply (homology and cohomology) Leray-Hirsch with r = 1.



This is most frequently applied when E is a vector bundle and E0 = E n 0, reduced to obtain a ball bundle as follows. Each ber Eb  = Rn admits a strong deformation n retraction onto the closed n-ball B . The (strictly) positive reals R+ act on the bers by scalar multiplication, and the quotient (or orbit space) of (E0 )b  = Rn n 0 n n 1  by (or under) this action is S = @ B . The Thom space of the vector bundle is the associated ber bundle with bers B n= Sn 1, the one-point compacti cations of the original vector-space bers.

Ex

1. If a Thom class u exists, then the ball bundle is H  -orientable. 2. One can now re-obtain the Gysin sequence for bers S0. From H  (RP n ; Z2)  = Z2[w]=(wn+1 ) ; where w 2 H 1 (RP n ; Z2 ) is a generator, it follows that H  (RP 1 ; Z2)  = Z2[w] : 3. Relate the Euler class and the Thom class. 4. Consider twisted versions of the Euler and Thom classes. 5. Let MO(n) denote the Thom space of the universal O(n)-bundle EO(n) ! ! BO(n). The inclusion BO(n) ,! BO(n + 1) induces a map MO(n) ^ S1 ! MO(n + 1) which yields a spectrum MO (cf. section 6.2 for spectral theories). The associated (unreduced) spectral homology theory MO is bordism, already encountered in section 5.8, and the cohomology theory MO is cobordism. 6. The cobordism ring is the graded ring  (MO) = N . Classically, it was described by saying that two compact n-manifolds are cobordant if and only if there exists a compact (n +1)-manifold with boundary, having precisely the two n-manifolds as its boundary. Equivalence classes of (compact) manifolds are the elements of the ring, with addition induced by disjoint union and multiplication by Cartesian product. The identi cation as homotopy rings is due to Thom [107].

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7. Similarly, there are Thom spectra MSO, MSpin, MU , and MSp. One says that MO is unoriented cobordism, and that the others are oriented, spin, unitary, and symplectic cobordism, respectively. There are other cobordism theories as well: the standard reference still seems to be Stong [103]. The introduction in Vershinin [113] may serve as a guide to more recent work.

7.3.6 Zeeman comparison theorem It is often useful to compare spectral sequences as we did above in the Leray-Hirsch theorem. Theorem 7.3.12 (Zeeman comparison) If rpq : Epqr ! Epqr is such that: r = E r = 0 for p; q < 0; (i) Epq pq 2 = E 2 E 2 , and the same for E 2 and 2 ; (ii) Epq pq pq p0 0q 1 = Epq 1 = 0 for (p; q) 6= (0; 0); (iii) Epq

(iv) 2p 0 is an isomorphism for p  0; then 20 q is an isomorphism for all q.



The proof is straightforward by downward induction on r; we refer to Switzer [106] for details.

Ex Write down the version where (iv) and the conclusion are interchanged, and both versions for cohomology.

7.4 Characteristic classes The idea here is that we would like to have (co)homology classes which are characteristic for a ber bundle, in the sense that they depend only on the fact that it is a G-bundle rather than any particulars. Recall that every G-bundle with model ber F over X is the pullback bundle of the universal bundle EG[F ] over BG by a map f : X ! BG, and that the isomorphism classes of bundles correspond to homotopy classes of such classifying maps; see pages 221. Now a map f : X ! BG induces a map on cohomology f  : h (BG) ! h (X ) for any cohomology theory h , and f  depends only on [f ] 2 [X; BG].

De nition 7.4.1 Elements of h(BG) for any cohomology theory h are called universal characteristic classes for G-bundles. If f : X ! BG is the classifying map for a G-bundle E over X , then the images in h (X ) of the universal classes under f  are called the characteristic classes of the bundle E . If X is a manifold

and u is any universal characteristic class for vector bundles, we write u(X ) as a shorthand for u(TX ).

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249

Usually, one selects generators of h (BG) according to some preference system and refers to them as the universal classes. The preference system employed is frequently an algebraic criterion for simplicity of the set of generators, such as minimality or computational ease. Thus in order to show that two bundles are not isomorphic, one seeks a cohomology theory for which the bundles have diering characteristic classes. It turns out that these classes also appear frequently as obstructions in existence problems, as we shall see subsequently (cf. Chapter 9). We begin by determining something about the structure of h (BG), assuming we already know something about the structure of h (G).

Theorem 7.4.2 (Borel) If h(G) = (x1 ; x2 ; x3; : : :) is an exterior R-algebra of nite type on generators of odd degree, then h (BG)  = R[y1 ; y2; y3 ; : : :] is a polyno-

mial R-algebra of nite type on generators of even degree, and deg yi = deg xi + 1. Moreover, the xi can be chosen so that yi =  (xi ) where  is the transgression.

The proof is a marvelous application of the cohomology version of the Zeeman Comparison Theorem just above. It concocts an abstract spectral sequence which ought to be the Serre spectral sequence of the universal bundle and which has the requisite structure, then compares it to the real Serre spectral sequence and concludes the two are isomorphic. We refer to Zeeman's original proof [125] for more details. Of course, this theorem is motivated by our results on the structure of the cohomology algebra of a nite-dimensional H-group, that for suitable ordinary theories it ts the hypotheses here exactly (see Theorem 5.11.4). In particular, this is the case for any Lie group. We now give some examples of H (BG; R) for selected classical Lie groups G. As this is just what we shall need, we omit the explicit computation of the exterior R-algebras H  (G; R) and refer you to [12] or [102], for example, for some details.

Ex of characteristic classes 1. H (BO; Z2)  = Z2[w1 ; w2 ; w3 ; : : : ] with deg wi = i for all integers i  1. These are the universal Stiefel-Whitney classes wi . Show that the inclusion  : BO(n) ,! BO induces  : H  (BO; Z2) ! ! H  (BO(n); Z2) = Z2[w1 ; : : : ; wn ]. 2. H (BL; Z2)  = Z2[s1 ; t1 ; w2 ; w3 ; : : : ] with the degree equal to the subscript

again. Here, BL denotes the classifying space of the stable Lorentz group L = lim! O(1; n). The identity component of O(1; n) has SO(n) as a deformation retract, inducing a map BL ! BO along which the wi are pullbacks of universal Stiefel-Whitney classes and the universal Stiefel-Whitney w1 pulls back to s1 + t1 . 3. H (BSO; Z2)  = Z2[w2 ; w3 ; : : : ] where the wi here may be identi ed with the universal Stiefel-Whitney classes. Indeed, SO ,! O induces a double cover p : BSO ! ! BO with p wi = wi for i  2 and p w1 = 0.

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CHAPTER 7. BUNDLE THEORY

4. H (BSpin; Z2)  = Z2[w4 ; w6 ; w7 ; w8 ; w10 ; : : : ] with deg wi = i 6= 2r + 1. Here, the ses 1 ! O(1) ! Spin(n) ! SO(n) ! 1 induces a bration BO(1) ,! BSpin ! ! BSO. The form of H  (BSpin; Z2) now follows from a spectral sequence argument and the fact that BO(1)  = RP 1 . = B Z2  5. H (BU ; Z)  = Z[c1; c2 ; c3 ; : : : ] with deg ci = 2i for i  1. These are the universal Chern classes ci . Study the inclusion U (n) ,! U as in Ex 1 above. Show that O = Re(U ) induces a map BO ! BU under which the mod 2 reduction of the pullback of ci is w2 i. The usual embedding

A B U (n) ,! O(2n) : A + iB 7 ! B A

also induces a map BU ! BO. What is the relation between the wi and ci now? 6. H (BSp; Z)  = Z[p1; p2 ; p3 ; : : : ] with deg pi = 4i for all integers i  1. These are the universal Pontrjagin classes pi . There are maps BO ! BU ! BSp under which pi pulls back rst to c2i and then to w4i mod 2. At this point, most readers will probably want to skip to the next chapter. Those who are seeking additional challenges may continue with the next, nal section of this chapter.

7.5 Nonabelian cohomology Here is one attempt to construct an ordinary cohomology H 2 with nonabelian groups as coecients, mostly based on the work of Dedecker [25]. The main objective is to obtain an interpretation similar to that of representatives of elements of H 1 (X ; G) as principal G-bundles. Let X be a paracompact Hausdor space (usually a smooth manifold). In addition to a sheaf of groups G over X we shall also need a subsheaf of Aut(G ) which contains the inner automorphisms; it will be denoted by A. We consider A as a sheaf of operators on G . More generally, A may be any sheaf of operators on G together with a map  : G ! A such that A1. (G ) are the inner automorphisms; A2. ((g)) = (g) 1 ;  2 Ax and g 2 Gx .

Ex Investigate the case when G is a normal subsheaf of H and A = Inn(H), the inner automorphisms. The triple  = (G ; ; A) is called a system of coecients. The standard de nition will apply: H 0 (X ; ) = H 0 (X ; G ) = (G ) :

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251

A 1-cochain is a pair of local sections (g; ) = (gij ; ij ) with respect to a covering U = Ui with Uij = Ui \ Uj 6= ? and

gij 2 (Uij ; G ); ij 2 (Uij ; A) such that ij ji = ii = 1 and gij = ij (gji1 ) and gii = 1. These are usually

called alternated cochains, but we shall not need any more general. We denote by Ca1 (U ; ) the set of 1-cochains.

Lemma 7.5.1 (Cochain groupoid) If we de ne a product on Ca1(U ; ) by (g0 ; 0 )
(g; ) = (g0 g; ) , 0 = (g); then Ca1(U ; ) forms a groupoid. For the proof of this and the next several results see Dedecker [25]. For a groupoid approach to topology see Brown [17], and for the view of a groupoid as a category with invertible arrows see Higgins [42]. A 2-cochain is a triple of local sections (; ; 0 ) = (ij ; ijk ; 0ij ) such that:

ij jk ki ij ji

kij

ijk

= ( ijk )0ij 0jk 0ki ; = ii = 1; = ki ( ijk ) or ijk = ik ( kij ); = ikj1 :

In a similar way to the case of 1-cochains, we de ne the product ( ij ; ijk ; ij0 )
(ij ; ijk ; 0ij ) = ( ij ; ijk ijk ; ij ) , ij0 = aij ; then the 2-cochains from a groupoid Ca2 (U ; ). Here of course ijk 2 (Uijk ; G ) where Ui \ Uj \ Uk 6= ?, etc. The boundary operator 1 : Ca1 (U ; ) ! Ca2(U ; ) is given by

1 (gij ; ij ) = ((gij )ij ; gij ij (gjk )ij jk (gkl ); ij ) or: 1 (g; ) = ((g);  g; ) for short. It is easy to verify that 1 maps cochains as indicated, and is in fact a groupoid homomorphism. A cochain is fundamental if and only if (g; ) = (g; 1) (; ; 0 ) = (; ; 1)

[ij (x) = 1 2 Ax ] [ij (x) = 1 2 Ax ]

For fundamental cochains we drop the right 1 and just write g and (; ) respectively. We denote the sets of fundamental cochains by Ca1 (U ; ) = Ca1 (U ; G ) and Ca2 (U ; ), respectively. Writing g = gij gjk gki ; 1 g = ((g); g) and clearly the cohomology of a fundamental cochain is fundamental. Also, left multiplication by arbitrary cochains preserves fundamentality.

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We say that g 2 Ca1 (U ; G ) is a 1-cocycle if and only if (in Uijk ) gik = gij gjk . Also, (; ) 2 Ca2 (U ; ) is a 2-cocycle if and only if (in Uijkl ) ij ( ljk ) = ilj ijk ikl : We denote the cocycles by Za1 (U ; G ) and Za2 (U ; ), respectively. Observe that g 2 Za1 (U ; G ) = Za1 (U ; ) if and only if 1 g = ((g); 1). One can de ne a coboundary 2 and obtain a similar statement about Za2 (U ; ), but we shall not need this. Via a straightforward calculation, we arrive at the following construction.

Lemma 7.5.2 (Groupoid of left operators) 1 : Ca1 (U ; G ) ! Za2 (U ; ) : If we de ne g
x = (1 g)x, then Ca1 (U ; ) becomes a groupoid of left operators on Ca2 (U ; ). Similarly, we de ne left operations of C 0 (U ; G ) and C 0 (U ; A) on Ca1 (U ; G ) and Ca2 (U ; ) via h
gij = hi gij hj 1 h
(; ) = ((hi )ij (hj 1 ); (hi ) ijk ): Note that only C 0 (U ; G ) operates on Ca1 (U ; G ); to obtain the operation of C 0 (U ; A) on Ca2 (U ; ), make the obvious change of variables. One may verify that 1 (h
g) = h
1 g h
: Za2 ! Za2 : Lemma 7.5.3 (Map on cocycles) Let h 2 C 0 (U ; G ) or C 0(U ; A), take (g; ) 2 Ca1 (U ; ), and (0 ; ) 2 Ca2 (U ; ). If either side is de ned then so is the other, and h
(1 (g; )(0 ; )) = 1 (h
(g; ))(h
(0 ; )): Thus, when h
is de ned, it maps 2-cocycles to 2-cocycles. Ex Why is it clear that h
maps 1-cocycles to 1-cocycles? Two 1-cocycles g and g0 are equivalent if and only if, for some h; k 2 C 0 (U ; G ), we have g0 = h
((0 k)g) : Two 2-cocycles (; ) and (0 ; 0 ) are equivalent if and only if for some (g; ) 2 Ca1 (U ; ) and h 2 C 0 (U ; A), (0 ; 0 ) = h
(1 (g; )(; )) :

Observe that we must have =  for the product on the right to be de ned whence 0 = h
(g). These are readily veri ed to be equivalence relations and

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253

equivalence classes are called cohomology classes. We denote the sets by H 1 (U ; G ) and H 1 (U ; ), respectively. Both have null or trivial classes, represented by h
0 k and h
1 (g; 1), respectively. In addition, H 2 (U ; ) has neutral classes represented by (; 1). As is usual in C ech-type constructions, we now de ne n H n (X ; ) = lim !U H (U ; ); n = 1; 2:

We usually write H 1 (X ; G ), as is traditional.  H ! 1 is a short exact sequence of sheaves Suppose next that 1 ! N ! G ! and de ne coecient systems  = (G ; ; A) given, 0 = (N ; ; A) induced, 00 = (H; 00 ; A=(N )) ; with 00 de ned via commutativity in G 1 1 N H /

/

/



1

00



(N ) 

/

/

A /



A00 

/

/

1

We now de ne connecting maps. Let h 2 H 0 (X ; H) = (X ; H) and U be a covering of X . This induces local representatives hi : Ui ! H. We have the easily-veri ed exact sequence 1 ! H 0 (X ; N ) ! H 0 (X ; G ) ! H 0 (X ; H) : By exactness, then, we can pull the hi back to gi 2 (Ui ; G ) and, for U \ Uj 6= ?, de ne unique nij 2 (Uij ; N ) by gj = gi nij on Ui : Then nij is a 1-cocycle, hence represents an element in H 1 (X ; N ). We can now construct
: H 0 (X ; H) ! H 1 (X ; 0 ) = H 1 (X ; N ) : h 7! n :

Ex Suppose that G is a constant sheaf, so we consider a ses 1 ! N ! G ! H ! 1 of groups. Then h : X ! H , and G bers over H as a principal N -bundle. Thus, h G is a principal N -bundle over X , and
h = [h G]. This depends on showing that H 1 (X ; G)  = f iso-classes of principal G-bundles over X g. Classically, this follows from the fact that the 1-cocycle condition is the transition function condition, and a computation to check the equivalence relations.

Now, [hij ] 2 H 1 (U ; H), so hij 2 Za1 (U ; H). Re ning U if necessary, we may assume that hij is the image of some gij 2 Ca1 (U ; G ) since the map H 1 (X ; G ) ! H 1 (X ; H)

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254

is epic. Set ij = (gij ) and observe that (; g) 2 Za2 (U ; ). It is now easy to check that [(; g)] depends only on [h] thus de ning
: H 1 (X ; H) ! H 2 (X ; 0 ):

Theorem 7.5.4 (Dedecker) This sequence is exact: 1 ! H 0 (X ; N ) ! H 0 (X ; G ) ! H 0 (X ; H)
! H 1 (X ; N ) ! H 1 (X ; G ) ! H 1 (X ; H)
! H 2 (X ; 0 ) ! H 2 (X ; ) ! H 2 (X ; 00 ); where we use all neutral classes 00 (including the null class) as the distinguished sets

in H 2 (X ; 0 ) and H 2 (X ;  ), and only the null class in H 2 (X ; ).

In order to obtain a geometrical interpretation of classes in H 2 , we shall make comparisons with Duskin [35]. Recall that an aspherical 1-truncated simplicial object Y over X consists of sets and maps

Y0 = fverticesg

Y1 O

Y1 = fdirected edgesg

d0 d1 s0 /

/

! Y0

/

X

d0 s0 = d1 s0 = 1Y0

Y1 (d0 ;d1 ) Y0  Y0 Y0 X /

/

Let

0 x f1 x j x ; x ; x 2 Y ; f ; f 2 Y g : 2 = fx0 f! 1 0 1 2 0 0 1 1 2 A 2-dimensional Kan action of G on Y (on the right) is given by a map 0 x f1 x ; g ) 7 ! [(f ; f ) g : x ! x ] 2 : 2  G ! Y1 : (x0 f! 1 0 1 0 1 2

such that

K 1: K 2:

(f; f ) 1 = s0 (x0 ); (s0 (x1 ); f ) 1 = f ((f23 ; f13) g0 ; (f23 ; f03 ) g1 ) g1 1g2 g0 = (f13 ; f03 ) g2

x1 B BB | | BBf13 ? || BB | | ? B || f 03 x0 B x3 BB | | BB | ||f23 ? BB | B || >

O

/

>

x2

The action is principal if and only if given f2 : x0 ! x1 there exists a unique g 2 G such that f2 = (f0 ; f1 ) g. Now, for us, let U be a covering of X and denote

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255

by X0 (vertices) the point-set fi j Ui 2 Ug and by X1 (directed edges) the point-set fi ! j j Ui \ Uj 6= ?g. We shall consider only the constant sheaf for a group

G. In order to obtain a suitable set of directed edges for a 2-dimensional Kan action, form the groupoid G (X0 ; G) with connected vertices determined by X1 ; see Zassenhaus [124], p. 181, for details. Clearly, G (X0 ; G) and X0 form an aspherical 1-truncated simplicial object over X (the map X0 ! X is epic in the appropriate category). Regard a 2-cocycle (; ) = (ij ; ijk ) as composed of maps ij which act as transport from vertex j to vertex i and of maps ijk which are attached at vertex i and live on the oriented 2-simplex (ijk). De ne 2 : 2  G ! G (X0 ; G) : ijk 7! 2 ( ijk ) as follows: 2 ( ijk ) = (Ujk ; ijk ) 2 G (X0 ; G): Then the cocycle condition

lki lij ljk = lk ( kij ) together with the other conditions of the de nition show that 2 is a principal 2-dimensional Kan action of G on G (X0 ; G). (Note that the cocycle condition provided K2 and our de nition of a 2-cochain (page 251) covers K1.) Modulo the equivalence relation, this establishes for H 2 the conjectural remarks on pp. 279{81 of Johnstone [55] in our case. To complete this picture, one need only show now that connected components of the category of principal 2-dimensional Kan actions over X correspond with classes in H 2 bijectively. This is straightforward. For a more traditionally geometric picture, observe that, if we take geometric realizations, jG (X0 ; G)j ! ! jX 1 j is a principal G-bundle. We also have embeddings jX1 j ,! X as a 1-skeleton for the covering U and its nerve, regarding the nerve of U as a simplicial approximation to X . From above, it follows that jG (X0 ; G)j together with the (realization of the) Kan action 2 : 2  G ! G (X0 ; G) determines a unique cohomology class in H 2 (X ; ); and, moreover, that every class is so determined. In Duskin's setting [35], the ijk form the 2-simplices of the K (G; 2)-torsor determined by G (X0 ; G). He was explicitly writing for abelian G, so the operators ij did not appear. Making appropriate modi cations, we could say that (; ) is a 2-simplex of a K (; 2)-torsor over X . Finally, we have a geometric interpretation of the second connecting map:
: H 1 (X ; H) ! H 2 (X ; 0 ) for constant sheaves. Hence we have the short exact sequence of groups 1 ! N ! G ! H ! 1: Let U be a covering of X and hij : Uij ! H represent a class [h] 2 H 1 (X ; H ). Recalling that G is a principal N -bundle over H , we would like to use the hij G to

256

CHAPTER 7. BUNDLE THEORY

construct a principal N -bundle over jX1 j. Re ning U if necessary, we may assume that each non-empty Ui \ Uj is contractible, as well as each Ui being contractible. In this case, it is obvious that the bundles hij G will de ne a principal N -bundle over any embedded realization jX1 j ,! X as desired. One now veri es readily that this bundle (together with the induced Kan action) represents
[h]. The connecting maps arising from short exact coecient sequences are traditionally known as Bockstein morphisms; cf. page 111.

Ex How successful was this attempt?

Chapter 8

Obstruction Theory Now wherefore stopp'st thou me?|Coleridge, Rime of the Ancient Mariner.

8.1 Preliminary ideas Perhaps the most elementary use of an algebraic entity to express the degree of absence of a property is in taking the kernel of a group homomorphism f , for ker f is the obstacle to f being injective. In this chapter we shall see how homotopy theory yields algebraic indicators for obstacles to extending and lifting maps. A simple example appeared at the beginning of Chapter 2, on page 25, where we observed that a continuous map f : Sn ! X extends to the (n + 1)-ball bounded by Sn precisely when f is null homotopic. So here the obstacle to extending f is precisely [f ] 2 n (X ).

Sn

o



f

/

fy

X

B n+1

fy

exists

() f  



|

Similarly, the obstacle to a lifting problem in a principal bration could be expressed as the homotopy class of the map into the classifying space, as we showed in Theorem 2.2.3. First we shall develop somewhat the cone extension result. Given f : X ! Y , then f   if and only if f extends to f y : CX ! Y where we view X ' X  f1g as a subspace of CX = X  I=X  f0g. The mapping cylinder of f is a homotopy equivalent of Y given by Mf = X  I [f Y = (X  I t Y )= where, for all x 2 X , the equivalence  identi es (x; 1) with f (x). The mapping cone of f is Cf = CX [f Y  = Mf =X  f0g : 257

CHAPTER 8. OBSTRUCTION THEORY

258

In fact the induced projection is a bration with ber type X :

X  f0g ,! Mf ! ! Cf : We shall want the following characterization for composites.

Proposition 8.1.1 Given X !f Y !g Z then gf is inessential if and only if g extends to gy : Cf ! Z . Proof: If gf   then 9 (gf )y : CX ! Z extending gf . Now we de ne  (y) for y 2 Y; gy : Cf ! Z : y 7 ! (ggf )y (y) for y = (x; t) 2 CX: Certainly gy jY = g and gy is well de ned since if y = f (x) 2 Y , then f (x) is

identi ed with (x; 1) in Cf and (gf )y (x; 1) = (gf )(x). Conversely, given (gf )y extending gf we de ne



y t 6= 0; F : X  I ! Z : (x; t) 7 ! (gf ) (x; t) for for t = 0;

and easily verify that this is a homotopy as desired.



In Chapter 3 we met the notion of attaching cells to a space on page 76. The cone construction increases the dimension by one; for example C Sn 1  = B n , where n 1 we grow out from each point of S a real interval [0; 1], then identify all copies of the nal point f1g. Our next result allows us to use nullhomotopy to detect the extensibility of a map from an (n 1)-skeleton to an n-skeleton. A worm has no earthy extension through the 2-skeleton of a mole's tunnels; the mole has exploited this for eons as a source of food. (It is not clear to what extent a worm comprehends when it begins to cross the 2-skeleton of a mole's tunnel complex; perhaps clever worms quickly reverse.) Later, we exploit this in an iterative approach to extensions.

Proposition 8.1.2 For any CW-complex X and each n, the n-skeleton is a mapping cone over the (n 1)-skeleton. Proof: For each n-cell en with attaching map  : (B n ; Sn 1) ! (X n; X n 1), we take a copy B n =  (en ) of the closed n-ball with its boundary Sn 1. Taking disjoint unions over  we obtain

n :

a

B n ;

a

`

Sn



1

! (X n ; X n 1) :

`

Then, as for individual balls, B n = C ( Sn 1). Moreover, if we set _ n = n j` Sn 1, then it unites with the skeleton inclusion and n to give a map

X n 1 [_ n

a n  B

! Xn :

8.1. PRELIMINARY IDEAS

259

This is a homeomorphism by construction and because the composite with identi cation, a  a 

Xn 1 t

is a closed map.

! X n 1 [_ n

B n

B n

! Xn ;



Corollary 8.1.3 For any CW-complex Z (in particular,  or I) and map h : X n 1  Z ! Y , suppose that we have maps h : B n  Z ! Y satisfying h (x; z ) = h( (x); z ) for x 2 Sn 1: Then there is unique solution hy to the extension problem Y Xn 1  Z h /


n:

Proof: By construction, the only possible non-trivial homotopy groups in X~ (n) are those above n, and these are isomorphic to those of X by jn .  We call X~ (n) the n-connected covering space of X ; it is usually denoted merely by X (n) .

Ex

1. Verify that X (n) is unique up to homotopy. 2. We compute X [n] for X = S1.

n = 0: X [0] = [r X0(r). Now X0(1) = (X [B 2 )= = B 2 , there being 1 generator of 1 (X ), and X0(r) = X0(2) for r > 1, since r (B 2 ) = 0 for r > 0. Thus X [0] = B 2 . n = 1: X [1] = [r X1(r). Now X1(1) = X since 2 (X ) = 0, and X1(r) = X1(2) = X for r > 1 since r (X ) = 0 for r > 1. Thus X [1] = X = S1. n > 1: X [n] = X = S1, since r (X ) = 0 for r > 1. Note that fg = X (1) is, up to homotopy, the simply connected covering space R of X = S1. In this case the Postnikov tower is short! X = S1 = X [1] X = S1 A 1 S1 = X [1] /

`

AA A i AAA 

B 2 = X [0] 

8.2. EILENBERG-MACLANE SPACES K (; N )

265

Moreover, the inclusion X ,! X [0] needs no extension so  = i. However, to reveal X [1] ! ! X [0] as a bration (up to homotopy) with ber K (1 (X ); 1) = S1, we can contract B 2 to a point and  to the constant map, giving:

K (1 (S1); 1) ,! S1 ! ! fg :

Theorem 8.2.6 There exist K (; n) of all possible types, unique up to homotopy

equivalence.

Proof: `All possible types' means: integer n  1 with  abelian when n > 1. The

construction depends on the following steps: 1. existence of M (; n); 2. homotopy groups above n can be killed; 3. functoriality ensures uniqueness up to homotopy equivalence. Construction of an M (; n) begins with the observation that if fSn j  2 Ag is a collection of disjoint n-spheres, then n (_ Sn) is the free (abelian if n > 1) group on a set of generators which bijectively corresponds to A; cf. pages 46 and 60. This gives M (; n) for free . If  is not free, then it can be resolved into a short exact sequence 0 ! R ! F !  ! 0 with R and F free groups. Then M (F; n) and M (R; n) are constructible and the factoring F=R  =  induces a cellular map f : M (R; n) ! M (F; n). This actually gives us M (; n) as the mapping cone Cf because we have a bration Mf ! ! Cf and hence, up to homotopy, another one

M (R; n) ,! M (F; n) ! ! M (; n) : The exact homotopy sequence gives the result since n 1 (M (R; n)) = 0 by construction. We leave it to the reader to check that M (; n) is unique up to homotopy equivalence. Now apply the Postnikov decomposition to obtain a relative CW-complex (M (; n)[n] ; M (; n)) with cells in dimensions above (n + 1) only and



  0 for r > n; r M (; n)[n] = r (M (; n)) for r  n:

Thus we have constructed K (; n) = M (; n)[n] .



Ex on K (; n) and Postnikov towers 1. X is weak homotopy equivalent to lim X [n] because the homotopy group functors commute with inverse limits.

CHAPTER 8. OBSTRUCTION THEORY

266

2. Each X [n] is obtained from X [n 1] by lling in its (n + 1)-holes so, up to homotopy, we can convert each projection into a bration K (n (X ); n) ,! X [n] ! ! X [n 1]; and if n (X ) = 0, then the bration has trivial ber. 3. The torus S1  S1 is a K (Z  Z; 1). The Klein bottle is a K (; 1) where  is the group with two generators a; b and one relation aba = b. Is any other surface a K (; n)? 4. RP 2 = M (Z2; 1) so K (Z2; 1) = (RP 2 )[1] ' RP 1 . C P 1  = S2 = M (Z; 2) so 1 [1] 1 K (Z; 2) = (C P ) ' C P . 5. Show BG ' K (; n) if and only if G ' K (; n 1); in particular, BG ' K (G; 1) if and only if G is discrete. 6. An f : X ! Y from a k-dimensional CW-complex is nullhomotopic if and only if k f is nullhomotopic where k : Y ,! Y [k] is from a Postnikov tower for Y . [Hint: CX n X has cells in dimensions below k + 2.] 7. (Hopf theorem) If X is a path-connected, nite CW-complex of dimension at most n, then H~ n (X ; Z)  = [X; Sn] for an ordinary H . 8. K (; n) is an H-space if and only if  is abelian. 9. K (; n) ' K (; n 1). 10. For each abelian group , we obtain a spectrum H with (H)n = K (; n). The associated spectral homology and cohomology theories H and H are just ordinary homology and cohomology with coecients . 11. If fn g is any possible sequence of homotopy groups, then it can be realized by some space. 12. Taking Postnikov sections determines functors

X [n]

X

] CW /



] : [f ] CW

[f [n]] /

Y [n]

Y





where f [n] : X [n] ! Y [n] is the solution of the extension problem

X /

n

f

Y 

n

f [n]



Y [n] 



/

X [n]

8.3. MOORE-POSTNIKOV DECOMPOSITION OF A FIBRATION

267

8.3 Moore-Postnikov decomposition of a bration We have set up a procedure for converting a space, up to homotopy, into a tower of brations with Eilenberg-Maclane bers. Our next move is to extend that procedure to the case of a bration. This will permit us to decompose a lifting problem into stepwise lifts while keeping a tight grip on the growth of homotopy complexity as the successive stages improve in approximation to the given bration. Given a bration F ,! E ! ! B we shall construct a sequence of brations.

K (n (F ); n) ,! E [n] ! ! E [n with E [0] = B and

i



E [n]

1]

   i (E ) for i  n;

= i (B ) for i > n: So, at the nth stage, E [n] `looks like' E from a lower-dimensional homotopy viewpoint and `looks like' B from above. In fact we shall obtain much more because of the way the above isomorphisms are induced and because compositions of the brations induce brations over B with bers forming a Postnikov tower of F . The construction depends on our previous procedure for killing homotopy groups and on the standard procedure for converting a map into a homotopy bration. We illustrate with a simple example. p Ex For F ,! E ! ! B , take S2 ,! S1  S2 ! ! S1. Then E [0] = S1. n = 1: Put X = E _ B = (S1  S2) _ S1. Killing 1 (B ) in X leaves S1  S2. Killing r (E ) also, for r > 1, leaves X 1 = S1. Hence we obtain the identity bration for E [1] ! ! E [0] as  ,! S1 ! !1 S1, with ber K (1 (F ); 1) = . n = 2: Killing r (B ) in X for r  2 leaves S1 S2. Killing r (E ) also, for r > 2, leaves X 2 = S1  K (2 (S2); 2). Hence we obtain a bration K (2 (F ); 2) ,! E [2] ! ! E [1] as K (Z; 2) ,! S1  K (Z; 2) ! ! S1. n = 3: Killing r (B ) in X for r  3 leaves S1  S2. Killing r (E ) in X for r > 3 leaves X 3 = S1  K (Z; 2)  K (Z; 3). Hence we obtain a bration K (3 (F ); 3) ,! E [3] ! ! E [2] as K (Z; 3) ,! S1  K (Z; 2)  K (Z; 3) ! ! S1  K (Z; 2). n = 4: Killing r (B ) in X for r  4 and r (E ) in X for r > 4 leaves X 4 = S1  K (Z; 2)  K (Z; 3)  K (4 (S2); 4), so giving a bration K (Z2; 4) ,! S1  K (Z; 2)  K (Z; 3)  K (4(S2); 4) ! ! S1  K (Z; 2)  K (Z; 3) :

And so on, with clear inclusions E ,! E [n] inducing qisomorphisms on homotopy n groups below the nth. In the brations F [n] ,! E [n] ! ! B , we see that the bers form a Postnikov tower of F = S2 since F [n] is S2 with all r (S2) killed for r > n.

CHAPTER 8. OBSTRUCTION THEORY

268

Moreover qn = p1  p2 


 pn induces isomorphisms on homotopy groups above the nth . This is trivially true in our example; indeed, q1 = 1 and q3 : S1  K (Z; 2)  K (Z; 3) ! S1 gives the zero isomorphism from 3+k (E [3] ) to 3+k (B ). The bration in this example is actually a trivial S2-bundle over S1 and at each stage of decomposition the conversion to a bration is trivial. However, it does allow a visualization of the development in complexity of E [n] with increasing n as successively more of the homotopy groups of E are admitted. We summarize the general situation as follows. Theorem 8.3.1 (Moore-Postnikov decomposition) Given a bration F ,! E p n ! B with connected base B then there exist brations E [n] p! E [n 1] for n  1 [ n ] and maps hn : E ! E with pn hn = hn 1 , such that: 1. E [0] = B and h0 = p; 2. the ber of pn has the homotopy type of K (n (F ); n); 3. hn induces isomorphisms of homotopy groups k (E )  = k (E [n] ) for k  n; 4. qn = p1 p2


pn : E [n] ! B induces isomorphisms k (E [n] )  = k (B ) for k > n;  5. letting F [n] denote the ber of qn , then hn jF : F ! F [n] is a Postnikov tower of F . See Figure 8.2 for a diagram.

Proof: We shall induct on n. Denote by E 1 the space resulting from E _ B after killing in it those groups corresponding to r (B ) for r  1 and to r (E ) for r > 1. We extend the map p _ 1B : E _ B ! B to give p~1 : E 1 ! E [0] = B , by sending new interior points to  2 B . We have a natural map h~ 1 : E ! E 1 induced by E _ fg ,! E _ B . Up to homotopy we can

replace p~1 by a bration with and

1 E [0] F 1 ,! E [1] p! !

E [1] = f(u; ) 2 E 1  B I j (0) = p~1 (u) and p1 (u; ) = (1)g

F 1 = f(u; ) 2 E 1  B I j (0) = p~1 (u) = (1) = g : But then F 1 is that remnant of the ber F in E which persists after killing homotopy groups above the rst. Therefore, up to homotopy, F 1 is K (1 (F ); 1). The map h~ 1 composes with the killing map

1E1   : E 1 ! E [1]

8.3. MOORE-POSTNIKOV DECOMPOSITION OF A FIBRATION

.. .

.. . pn+1

K (n (F ); n)

o ooo o o ooo

K (n (F ); n)

o ooo o o ooo

W



E [n] u 

7



F [n] u 

w

7

w

O

hn uu uu u u E& 6 && 66 && 66 && 66hn 1 pn K (n 1 (F ); n 66 && o 66 && ooo o 6 o && oo && [n 1] E && ** && ** && p * h1 & n 1 * * && ** && ** && ** . .. && ** && qn 1 ** && ** && ** p && 2 ** K (1 (F ); 1) && **oo o o & o * ** ooo [1] ** E CC ** CC ** CC C q1 =p1 CC ** CC **

h0n uuu

O

/

o

W

7





W

/

o

:



269

w

o





W

7







w

o



1)

:

u uu F& 6 && 66 && 66 && 66h0n 1 K (n 1 (F ); n 66 && o 66 && ooo o 6 o && oo && [n 1] F && && & h01 && && && && .. && . && && && K (1 (F ); 1) && o && o ooo & o o oo W

7





w



O

/





W

7



F [1] 



O

/

w



!

E [0] = B !

Moore-Postnikov tower of E

Postnikov tower of F

Figure 8.2: Moore-Postnikov decomposition of a bration

1)

CHAPTER 8. OBSTRUCTION THEORY

270 to yield

h1 : E ! E [1]

which evidently induces an isomorphism of rst homotopy groups. The map q1 = p1 : E [1] ! B : (u; ) 7! (1) induces isomorphisms of homotopy groups above the rst since p1 arises from the identity on B in E _ B and only B contributes to these homotopy groups in E [1] . Now suppose that we have found brations satisfying the requirements of the theorem up to n 1. Thus we have

E [n x

E2

hn 1xxxx x xx xx hn 2 /

1]

O

/ o


n. (Observe that killing r (E [n 1] ) for r  n leaves, up to homotopy, E _ Bn where Bn is B with homotopy groups killed up to the nth .) Thus we have

E

o



/

E_

o



/

1] kill

E _ E [n

xx xx x x p~n xx

hn 1 _1 

E [n 1] |



|

En x

E [n]

/

'

O

o

/

Fn

pn 

E [n 1] 

where pn is the bration version of the projection p~n induced by hn 1 _ 1, and a map hn : E ! E n is induced by the killing process. Clearly, hn induces isomorphisms on homotopy groups up to the nth . Also, pn induces isomorphisms above n, but qn 1 = p1 p2 :::pn 1 induces isomorphisms above (n 1) and so qn = qn 1 pn induces isomorphisms above n. The bration pn has ber F n = pn fg = f(u; ) 2 E n  (E [n 1] )I j (0) = p~n (u) = (1) = g

8.4. HOMOTOPY COFUNCTORS

271

which is that remnant of the ber F in E which persists after killing homotopy groups above the nth . Hence, n (F n ) = n (F ), and r (F n ) = 0 for r > n. But, by hypothesis, F n 1 ' K (n 1 (F ); n 1) and F n 1 is the remnant of F in E after killing homotopy groups above the (n 1)th , so r (F n ) = 0 for r < n 1. Thus in order to establish F n ' K (n (F ); n), it is required only to show that n 1 (F n ) = 0. Now, by the induction hypothesis, the bers F [r] of qr for r  n 1 form a Postnikov tower of F so n 1 (F n ) will contribute a copy of K (n 1 (F ); n 1) and F n 1 to F [n] ; but this will project by pn onto a similar copy in F [n 1] so pn fg will contain no such copy and hence n 1 (F n ) = 0. 

8.4 Homotopy cofunctors We indicated above that cohomology classes are useful for characterizing obstructions to extensions. More generally, we use homotopy cofunctors which all turn out to be representable, up to natural equivalence, by homotopy classes of maps into some standard classifying spaces.

g ! Set satisDe nition 8.4.1 A homotopy cofunctor is a cofunctor H : Top

fying the coequalizer and wedge properties: r] (i) if Y [! R is the coequalizer of

[f ] [g ]

X

/

/

Y

and H [f ]u = H [g]u, then there exists v 2 H (R) such that H [r]v = u; i _ X then we have an isomorphism (ii) if X !  

fH [i]g : H

_ !Y 

X =



H (X ) :

Proposition 8.4.2 (Suspension object groups) If H is a homotopy cofunctor, then it induces a group structure on each suspension object and it is abelian on double suspensions. If H is group-valued, then the two group structures coincide. Ex [SX; Y ] = [S1 ^ X; Y ] which is naturally equivalent to [S1; Y X ] = 1 (Y X ).

Hence, the cofunctor [ ; Y ] is group-valued on suspension objects. On double suspensions we use S (SX ) = S1 ^ S1 ^ X  = S2 ^ X giving [S (SX ); Y ]  = [S2; Y X ] = 2 (Y X ) which is abelian.

CHAPTER 8. OBSTRUCTION THEORY

272

Proof: Let SX be a suspension and denote the comultiplication on SX by ( ([2t; x]; ) t 2 [0; 21 ] : SX ! SX _ SX : [t; x] 7 ! (; [2t 1; x]) t 2 ( 21 ; 1] using SX = ([0; 1]=f0; 1g) ^ X ; cf. Section 2.3, page 41 in particular. Denote the projections from SX _ SX onto the two copies of SX by p; q respec-

tively. Then we obtain maps

SX /

SX _ SX (p;q) SX  SX : /

Applying H gives a multiplication

H (SX )  H (SX ) (H [p];H [q]) H (SX _ SX ) /

H [ ] /

H (SX ) :

Equivalently, (H [p]; H [q]) = (H [i]; H [j ]) 1 , where i and j are the inclusions of the copies of SX in SX _ SX . The H-cogroup structure of SX ensures that the multiplication on H (SX ) yields a group. Clearly, if H is group-valued then it yields homomorphisms on homotopy classes of maps, and in that case the two group structures are equivalent up to isomorphism. 

Corollary 8.4.3 For any homotopy cofunctor H , H (Sn) is a group for n  1 and abelian for n  2.  We call H (Sn) the nth coecient group of H . In particular the nth coecient group of H = [ ; Y ] is just n (Y ).

Ex 1. Show that H r ( ; G) is a homotopy cofunctor and that its nth coecient group is G if n = r  1, and trivial if n 6= r. 2. If H is a homotopy cofunctor and  is a singleton, then use _ =  to deduce that H () is also a singleton. A space Y is called a classifying space for a homotopy cofunctor H if for all CW-complexes X , H (X )  = [X; Y ] : Then every element u 2 H (Y ) determines a map Tu : [X; Y ] ! H (X ) : [f ] 7 ! H [f ]u : In particular, by substituting spheres Sn for X we can test the sensitivity of u 2 H (Y ) to homotopy information. We say that u is universal if Tu is an isomorphism for all n  1.

8.4. HOMOTOPY COFUNCTORS

273

Ex 1. If u 2 H (Y ) and u0 2 H (Y 0 ) are universal, then Y and Y 0 are weak homotopy equivalent (WHE). 2. For path connected Y; Y 0 and H = [ ; Y 0 ], u 2 H (Y ) is universal if and only if u is a weak homotopy equivalence.

Theorem 8.4.4 (Existence of classifying space) Every homotopy cofunctor H has a CW-complex Y which is a classifying space for H ; moreover, there exists some universal u 2 H (Y ) if and only if Y is a classifying space for H . Proof: We outline the construction; it resembles the synthesis of a K (; n). (i) Put Y0 = ; then H (Y0 ) is a singleton, fu0g say. (ii) Construct a bouquet of 1-spheres X1 = _2H (S1) S1. (iii) De ne Y1 = Y0 _ X1 ; then the wedge property yields u1 2 H (Y1 ) which restricts to u0 on Y0 and satis es H [i ]u1 =  for all  2 H (S1) with inclusions i Y . But then T [i ] =  so T  (Y ) = H (S1) and u is 1-universal. S1 ! 1 u1 1 1 u1  (iv) Construct a bouquet of 2-spheres X2 = _2H (S2) S2. (v) For each [f ] 2 1 (Y1 ) with H [f ]u1 = 0, ll in (by attaching via f ) a 2-cell in Y1 , to yield W1 . Put Y2 = W1 _ X2 . (vi) The wedge property yields u2 2 H (Y2 ) restricting to u1 on Y1  W1 and i Y . H [i]u2 =  for all  2 H (S2) with inclusions S2 ! 2 (vii) There is a commutative diagram with exact row

2 (Y2 ; Y1 )

@ /

i

1 (Y1 0: 8 0; x < 0; < (c) F = ff1g where f1 (x) = : 1; x = 0; 2; x > 0: (d) F = ff2g where f2 (x) = x for all x. (e) F = ff j f : R ! R is continuous with the usual topology on Rg. 61. Let (X; T ) be a topological space and let Cb (X; R) be the vector space of continuous bounded functions from X to R where R has its usual topology. Prove that the weak topology on X with respect to Cb (X; R) equals T if and only if (X; T ) is completely regular (that is, (8 x0 2 X , 8 open nbhd N of x0 ) 9 continuous f : X ! [0; 1] with f (x0 ) = 0 and f (x) = 1 if x 2 X n N ; cf. Hocking and Young [46], p. 74). 62. If (X; T ) is normal and E; F are nonempty disjoint closed subsets, then from each closed interval [a; b] there is a continuous map taking the values a on E and b on F . (Remember: normal ) can separate disjoint closed sets.) 63. If (X; T ) is T2 or T1 , then so is (X; T 0 ) if T  T 0 ; so also is any subspace. 64. The product space X  Y is T2 if and only if X and Y are. 65. Products preserve second-countability and separability. 66. (X; T ) is T2 if and only if the diagonal
= f(x; x) j x 2 X g is closed in X  X. 67. Given a space (X; T ) and a base B for T , (X; T ) is compact , every open cover by sets in B has a nite subcover. 68. (X; T ) is compact ) every collection of closed subsets with the nite intersection property has nonempty intersection. 69. X  Y is connected , X and Y are connected. 70. If T ; T 0 are topologies on X with T  T 0 then (X; T 0 ) is compact ) (X; T ) is compact. 71. Show that the space (R; T ) in 24 has a compact subset with noncompact closure. Can you generalize this? 72. (i) (fa; bg; Tdis ) is disconnected; (fa; bg; (?; fag; fa; bg)) is connected. (ii) Closures and continuous maps preserve connectedness. (iii) Unions (of sets that meet) preserve connectedness. (iv) X is connected if whenever X = A [ B for some nonempty A; B then either A \ B 6= ? or A \ B 6= ?.

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329

73. Given a compact Hausdor X and an open, proper subset U , then U 1 = U [ f1g has a topology such that there is a homeomorphism h : U 1 ! ! X=(X nU ) where h coincides with the natural projection p on U and h(1) = p(X n U ). Furthermore, if x 2 X then (X n fxg)1  = X . For example,

Sn n fnorth poleg  = In n @ In ; = B n n Sn 1  = Rn  so Sn  = In=@ In : = B n =Sn 1  = (Rn )1  [Look up one-point compacti cation for locally compact Hausdor spaces;

cf. [4], p. 73.] 74. If A is a connected subspace of X , then A  Y  A ) Y is connected. For example, Rn+1 n f0g, Sn, and RP n are connected for n  1. 75. Let A = f(x; y) 2 R2 j x = 0; y 2 [ 1; 1]g and B = f(x; y) 2 R2 j x 2 (0; 1]; y = cos(=x)g. (Sketch!) Show that (i) A and B are connected, (ii) A \ B = ?, and (iii) if X = A [ B then X is connected. (For (iii) consider X = U [ V for some U and V both open and closed in X and assume that for some a 2 A we have a 2 U .) [Observe that a connected space can consist of two parts that do not intersect but the converse is impossible!] 76. As for the previous one with

A = f(x; 0) 2 R2 j x 2 [ 21 ; 1]g B = f(x; x=n) 2 R2 j x 2 [0; 1]; n 2 N g : 77. Given (X; T ) put H (X ) = fcontinuous f : X ! Z2g, where Z2 = f 1; +1g with the discrete topology; then for all f; g 2 H (X ) de ne f + g : X ! Z2 : x 7! (f (x) + g(x)) (mod 2). Prove (i) f + g is continuous, 8 f; g 2 H (X ); (ii) (H (X ); +) is an Abelian group; (iii) (X; T ) is connected , (H (X ); +) is cyclic of order 2, that is isomorphic to Z2 viewed as a group. Also, nd for all k 2 N a space Xk such that H (Xk ) is isomorphic to k copies of Z2.

Appendix C

Manifolds and Bundles A topological space is the least structure that can support arguments concerning continuity and limits. A manifold is the least structure that can support arguments concerning dierentiability and tangents. A bundle is a generalization for manifolds of the concept of a product space; that is, a possibly twisted product. Dodson and Poston [31] provides an introductory treatment of manifolds and bundles, with applications in geometry and relativity. A short account of manifold geometry and calculus on manifolds is given in Dodson [30]. Comprehensive treatments are given in Spivak [99], Kobayashi and Nomizu [61, 62], and O'Neill [86].

C.1 Manifolds A smooth n-manifold M is a Hausdor topological space together with a collection of maps (the charts)

f : U ! Rn j  2 Ag from open subsets U of M , which satisfy: i) fU j  2 Ag is an open cover of M ; ii) each  is a homeomorphism onto its image; iii) whenever U \ U 6= ?, then the maps between subsets of Rn

   :  (U \ U ) !  (U \ U ) ;    :  (U \ U ) !  (U \ U ) ; have continuous derivatives of all orders (are C 1 or smooth). We call f(U ;  ) j  2 Ag an atlas of charts for M ; the properties of M are not

signi cantly changed by adding more charts. The terminology is borrowed from 331

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APPENDIX C. MANIFOLDS AND BUNDLES

mapmaking, where an atlas is a collection (book) of charts. A topological manifold is similar, but without any dierentiability requirements. All our manifolds are smooth. Intuitively, an n-manifold consists of open subsets of Rn , the  (U ), pasted together in a smooth fashion according to the directions given by the    . For example, S1 with its usual structure can be presented as a 1-manifold by pasting together two open intervals, each like ( ; ). Similarly, S2 has an atlas consisting of two charts f(UN ; N ); (US ; S )g where UN consists of S2 with the north pole removed, US consists of S2 with the south pole removed, and the chart maps are stereographic projections. Thus, if S2 is the unit sphere in R3 centered at the origin then: N : S2 n fn:p:g ! R2 : (x; y; z ) 7 ! 1 +1 z (x; y) S : S2 n fs:p:g ! R2 : (x; y; z ) 7 ! 1 1 z (x; y) : Essentially the same types of maps work also for the higher dimensional spheres.

Ex 1. Rn has an atlas consisting of just one chart, the identity map. 2. Find another atlas for S2 consisting of projections of six hemispheres onto three perpendicular planes through the origin. Show that this atlas generalizes also for higher dimensional spheres. 3. Find atlases for the cylinder S1  (0; 1), and for the torus S1  S1. Show that the notion of products of charts generalizes for arbitrary product manifolds. It is not dicult to test whether a collection f(U;  ) j  2 Ag constitutes an atlas for a given space M , but it is altogether nontrivial to discover how many distinct manifold structures are possible. Some results for Rn and Sn are given in Table C.1. (Cf. Donaldson [33] for a discussion of the initial excitement at R4 .)

C.2 Tangent spaces From elementary analysis we know that the derivative of a function is a linear approximation to that function, at the chosen point. Thus, we need vector spaces to de ne linearity and these are automatically present at each point of Rn . At each point x of a manifold M we construct a vector space TxM , called the tangent space to M at x. For this we employ equivalence classes [T(x)Rn ] of tangent spaces to the images of x,  (x), under chart maps de ned at x. That is, we borrow the vector space structure from Rn via each chart (U ;  ) with x 2 U ,

C.2. TANGENT SPACES

333

n

Sn

Rn

1 1 1 2 1 1 3 1 1 1 1 4 5 1 1 6 1 1 7 28 1 8 2 1 8 1 9 10 6 1 11 992 1 1 1 12 13 3 1 2 1 14 15 16256 1

Table C.1: Numbers of distinct dierential structures on real n-space and n-spheres. then identify the borrowed copies. The result, for x 2 S2 embedded in R3 , is simply a vector space isomorphic to the tangent plane to S2 at x. Actually, the formal construction is independent of M being embedded in this way. However, the Whitney Embedding Theorem [120] says that an embedding of an n-manifold is always possible in R2n+1 . Once we have the tangent space Tx M for each x 2 M we can present it in coordinates, via a choice of chart, as a copy of Rn . The derivatives of the change of chart maps, like @ (   ) (x1 ; x2 ;


; xn ) ; i 

@x

provide linear transformations among the representations of Tx M . Next, we say that a map between manifolds

f :M !N is dierentiable at x 2 M , if for some charts (U; ) on M and (V; ) on N with x 2 U; f (x) 2 V , the map

 f jU   : (U ) ! (V ) is dierentiable as a map between subsets of Rn and Rm , if M is an n-manifold and

N is an m-manifold. This property turns out to be independent of the choices of charts, so we get a linear map Txf : TxM ! Tf (x)N :

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APPENDIX C. MANIFOLDS AND BUNDLES

Moreover, if we make a choice of charts then Txf appears in matrix form as the set of partial derivatives of  f   . The notation Tx f for the derivative of f at x is precise, but in many texts it may be found abbreviated to Df , f, f 0 or Tf , with or without reference to the point of application. When f is a curve in M , that is, a map from some interval

f : [0; 1] ! M : t 7! f (t) ; then Tt f is sometimes denoted by f_t . This is the tangent map to f at t and the result of its application to the standard unit vector to R at t, f_t (^1), is the tangent vector to f at t . It is quite common for this tangent vector also to be abbreviated to f_t . In a natural way we can provide a topology and dierential structure for the set of all tangent vectors in all tangent spaces to an n-manifold M :

TM =

[

x 2M

Tx M ;

details are given in [31]. So, it actually turns out that TM is a 2n-manifold, called the tangent bundle to M . For example, if M = S1 with the usual structure then TM is topologically (and as a manifold) equivalent to the in nite cylinder S1  R. In general, this simple situation is quite rare and it is rather a deep result that for spheres T Sn is equivalent to Sn  Rn only for n = 1; 3; 7 : For other spheres, their tangent bundles consist of twisted products of copies of Rn over Sn. In particular, T S2 is such a twisted product of S2 with one copy of R2 at each point.

Ex Why is it intuitively clear that the tangent bundle to the torus M = S1  S1 is simply S1  S1  R2 ? What about the case when M is a Klein bottle? Ex If we replace each TxM by its dual space TxM , of real-valued linear maps on TxM , we get the cotangent bundle T M . For further information, see Dodson [29], p. 100 et seq., Brickell and Clark [15], p. 116 et seq., and Husemoller [52], Chapter 15. The technical term for an n-manifold M that has a trivial product tangent bundle TM  = M  Rn is parallelizable and this property is discussed in the cited texts. A map f : M ! N between manifolds is just called dierentiable if it is differentiable at every point of M , and a dieomorphism if it is dierentiable with a dierentiable inverse; in the latter case M and N are said to be dieomorphic manifolds. Dieomorphism implies homeomorphism, but not conversely. For example, the sphere S2 is dieomorphic to an ellipsoid, but only homeomorphic to the surface of a cube because the latter is not a manifold (it has corners and sharp edges).

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335

C.3 Calculus on manifolds The formal development of calculus on manifolds is in one sense straightforward: we need only to take the usual operations locally in Rn and synthesize their appropriate equivalence classes under charges of charts that are admissible in the atlas. This ensures that we obtain intrinsic, coordinate-free de nitions. What is less obvious is that exterior calculus turns out to be the correct generalization of ordinary vector calculus. The fundamental operation of exterior dierentiation, in the presence of a metric, actually recovers all of vector calculus in R3 and extends it to arbitrary dimensional manifolds in an amazingly powerful way. An m-form is a purely antisymmetric, real-valued, multilinear function on an argument of m tangent vectors, de ned smoothly over the manifold. A 0-form is a real valued function on the manifold. Thus, the space 0 M of 0-forms on M consists of sections of the trivial bundle M  R. The space 1M of 1-forms on M consists of sections of the cotangent bundle T M , and k M consists of sections of the antisymmetrized tensor product of k copies of T M . Locally, a 1-form has the local coordinates of an n-vector, a 2-form has the local coordinates of an antisymmetric n  n matrix, etc. Quite generally,
a k-form on an n-manifold has nk independent local coordinates. It follows that the only k-forms for k > n are the zero k-forms. The technicalities involved are not dicult but they are somewhat intricate and there are some subtleties. We shall summarize below the main algebraic de nitions for tensor and exterior algebra and calculus. In transcribing problems from the physics literature, it is important to recognize that not always is it clear whether reported formul are valid only locally in some chart, or globally; sometimes the underlying manifold is not mentioned either. Also, care needs to be taken to avoid confusion among the various (in nite-dimensional vector) spaces of sections of bundles (like k M ), the bundles themselves which are pairs of manifolds (like TM ! ! M ), and the local vector and tensor spaces at a point x 2 M constructed from copies of TxM and TxM .

C.3.1 Summary of formul

We shall suppose always our vector spaces are over R. There are three fundamental operations on nite-dimensional vector spaces (in addition to taking duals): direct sum , tensor product , and exterior product ^ on a space with itself. Let F; G be two vector spaces, of dimensions n; m respectively. Take any bases fb1;


; bng for F; fc1 ;


; cm g for G, then we can obtain bases

fb1;


; bn ; c1 ;


; cm g for F  G ; fbi cj j i = 1;


; n; j = 1;


; mg for F G ; fbi ^ bj = bi bj bj bi j i = 1;


; n; i < j g for F ^ F : So, F  G is essentially the disjoint union of F and G with their zero vectors identi ed. In a formal sense (cf. Dodson and Poston [31], p. 104), F G can be viewed as the vector space L(F  ; G) of linear maps from the dual space F  = L(F; R)

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APPENDIX C. MANIFOLDS AND BUNDLES

to G. Recall also the natural equivalence (F  )  = F . By taking the antisymmetric part of F F we obtain F ^ F . We deduce immediately: dim F  G = dim F + dim G ; dim F G = dim F
dim G ; dim F ^ F = 21 dim F (dim F 1) :

Observe that only for dim F = 3 can we have dim F = dim(F ^ F ). Actually, this is the reason for the existence of the vector cross product  on R3 only, giving the uniquely important isomorphism

R3 ^ R3 ! R3 : x ^ y 7 ! x  y and its consequences for geometry and vector calculus on R3 . Each of the operations ; and ^ induces corresponding operations on linear maps between spaces. Indeed, the operations are thoroughly universal and categorical, so they should and do behave well in linear algebraic contexts. Brie y, suppose that we have linear maps f; h 2 L(F; J ) g 2 (G; K ) then the induced linear maps in L(F  G; J  K ), L(F G; J K ) and L(F ^ F; J ^ J ) are

f  g : x  y 7 ! f (x)  g(y) ; f g : x y 7 ! f (x) g(y) ; f ^ h : x ^ y 7 ! f (x) ^ h(y) : It is clear how to iterate  and , but not so easy to visualize F ^ F ^ F ^ F = A4 F , for example. Recall that our eventual interest is in applying our operations to tangent spaces, smoothly over a manifold, and in particular using ^ on each cotangent space Tx M . So consider the case F = Tx M = L(TxM; R) and put k factors z }| { k Ax M = Tx M ^ TxM ^


^ Tx M ; k factors }| { z Tk Mx = Tx M TxM


TxM : Explicitly, the alternating operator is given by

Ak : Tk Mx ! Akx M : w 7 ! wA where the (real-valued) k-linear map w 2 L(Tx M  TxM 


 Tx M; R) is sent to the alternating k-linear map

X

wA : (v1 ; : : : ; vk ) 7 ! k1! sgn( )w(v (1) ; v (2) ; : : : ; v (k) )  2Sk

C.3. CALCULUS ON MANIFOLDS

337

with summation over the symmetric group Sk of all permutations of the indices f1; 2; : : :; kg. We can also use the alternating operator to de ne (Ak V ) ^ (Am V ) = Am+k V = Am+k (Ak V Am V ) : The vector spaces Akx M for x 2 M can be tted smoothly together
over M n so the to yield a bundle Ak M with ber Ak (Rn ) ; this ber has dimension k
n k total space of the bundle, A M , is a manifold of dimension n + k . We denote by k M the space of sections of the natural bundle surjection Ak M ! ! M , but others frequently use k M for the total space as well. By tting together the tensor product of k copies of Tx M and m copies of TxM , smoothly over the manifold M , we obtain the bundle of (k; m)-tensors Tkm M ! ! M, with ber the corresponding tensor product of copies of Rn and (Rn ) . Full details can be found in Dodson and Poston [31], but at each x 2 M an element of TkmM will simply have the appearance of an (m + k)-linear map, from m copies of TxM and k copies of TxM to R. So, given a choice of basis for the tangent space TxM , it induces a dual basis for TxM and consequently a basis for (Tkm M )x. Elements of the latter then appear in coordinate form as nm+k arrays; those of Akx M as purely antisymmetric nk arrays. It happens that the very nature of the induction process for the tangent spaces, directly from the choice of the dierentiable structure for the manifold, actually gives a de nite role to tangent vectors. An element v 2 TxM turns out to be a derivation on smooth real functions de ned near x 2 M . In a chart about x, v is expressible as a linear combination of the partial derivatives with respect to the chart coordinates x1 ; x2 ; : : : ; xn as

v = v 1 @1 + v 2 @2 +


+ v n @n with @i = @x@ i , for some vi 2 R. This is often abbreviated to v = vi @i , where summation is to be understood over repeated upper and lower indices, the summation convention of Einstein. The dual base to f@i g is written fdxi g and de ned by

dxj (@i ) = ij =

 1 if i = j ; 0 if i 6= j :

So  2 TxM = A1x M is expressible as

 = 1 dx1 + 2 dx2 +


+ n dxn = i dxi for some i 2 R, but 2 A2x M as

=

X i 0 of the form:  : U  ( ; ) ! M : (x; t) 7 ! (x; t) which satis es for all x 2 U

(i) (x; 0) = x; (ii) cx : ( ; ) ! M : t 7! (x; t) is a curve with tangent vector c_x(t) = v  cx(t). Such a map  is called a (local) ow for the vector eld v. Intuitively, we can think of v as a eld of little arrows (like a west wind on the Earth) and  as a family of curves which join up the arrows. Each curve cx is called an integral curve of v through x. We can write the (local) ow  in the form of a map t : U ! M : x 7 ! (x; t) for each t 2 ( ; ). Then it can be shown that t  s = t+s = s  t whenever all are de ned on U . Actually, we have fudged a bit:  might depend on x, which is the idea behind local in local ow; if it does not, we call the ow global. When all the intervals ( ; ) = R, the ow is called complete Given two vector elds u; v on M their commutator or Lie bracket is the new vector eld [u; v] de ned as the derivation on real functions f by [u; v](f ) = u(v(f )) v(u(f )) : Locally, if u = ui @i and v = vj @j , we nd that [u; v] = (ui @i vj vi @i uj )@j : Now let and  denote local ows for u and v on U , respectively, giving for suitable s; t, s : U ! M : x 7 ! (x; s) ; t : U ! M : x 7 ! (x; t) :

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339

Then it turns out that the two ows commute if and only if the commutator of the vector elds is zero: t  s = s  t , [u; v]jU = 0 : We want one further generalization for our k-forms. In some applications, they need to take values not in R but in some other vector space V , which may vary smoothly over the manifold. For example, a tangent space-valued k-form would be a section of the tensor product bundle

Ak M M TM ! !M where M denotes the result of smoothly forming Akx M TxM over x 2 M . These

are called vector-valued forms, and one similarly considers such things as Lie algebra-valued forms. We consider next exterior calculus, the generalization of vector calculus. The exterior derivative is a linear map on k-forms satisfying

(i) d : k M ! k+1M ( d has degree +1); (ii) df = grad f if f 2 0M (locally, df = @if dxi ); (iii) if  2 aM and 2 M , then d( ^ ) = d ^ + ( 1)a  ^ d ; (iv) d2 = 0. In fact, this d is unique in satisfying these properties. A k-form  2 k M is called: closed if d = 0 2 k+1 M; so  2 ker djk M exact if  = d for some 2 k 1 M; so  2 im djk 1 M (or if k = 0;  is constant) locally exact if for all x 2 M; there is a neighborhood of x on which the restriction of  is exact. It follows easily from d2 = 0 that exact =) locally exact =) closed, but with more diculty that closed =) locally exact (cf. Bishop and Goldberg [10]). Appendix E outlines the use of a computer algebra package for exterior calculus. A volume form on an n-manifold M is a nowhere-zero n-form  2 n M . Such a form actually exists if M has an underlying Hausdor topology with a countable base, and M is orientable. Now, orientability is a topological property but it is

340

APPENDIX C. MANIFOLDS AND BUNDLES

conveniently detectable for a manifold M if M admits an atlas for which all the change of chart maps   1 : (xi ) 7 ! (yj ) have a positive Jacobian determinant

 @yi 

J = det @xj : For, the standard volume form on Rn with respect to rectangular coordinates (xi ) is dx1 ^ dx2 ^


^ dxn . Also, it is not dicult to see that if we transform to any other coordinates (yj ) then the two volume forms are related by:

dy1 ^ dy2 ^


^ dyn = J dx1 ^ dx2 ^


^ dxn : Given a volume form  on M , it is possible to de ne an integral of an n-form ! on M if the n-form has compact support: that is, is zero outside a compact subset. The trick here is to use each chart map  : U ! Rn to transfer the integration locally

to Rn , and then share out the contributions from overlapping charts by means of the following device. A partition of unity on M is a collection of non-zero functions fp : M ! R j  2 Ag, with M n p (0)  U for some chart (U ;  ), and for all x 2 M

X

2A

p (x) = 1 :

This sum actually will have only a nite number of nonzero terms because, for our manifolds, each point meets only nitely many charts. A partition of unity like this, which is subordinate to an atlas, is the main device for fairly sharing the contributions of dierent charts in a variety of constructions on a manifold. Further discussion can be found in Brickell and Clark [15]. For a smooth manifold, paracompactness (cf. x B.4) is equivalent to the existence of a smooth partition of unity subordinate to the atlas. Now we can convert an integral of an n-form on M into an appropriate combination of multiple integrals on Rn and use Fubini's Theorem to express the latter as iterated integrals. There is a detailed discussion of the technicalities in Bishop and Goldberg [10], Chapter 4. In particular, if  is a volume form on M then we de ne the -volume of M to be the integral of  over M : vol M =

Z

M

:

Full exploitation of integration of forms on manifolds requires us to introduce the concept of a manifold with boundary. This simply requires us to relax our condition on charts from being homeomorphs of Rn to being homeomorphs of half spaces in Rn , like f(x; y) j x  0g in R2 . This extends our concept of a manifold to include spaces like a closed disk or a nite cylinder, for example. The boundary @M of a manifold M is the set of points coordinatized by points on the boundaries

C.3. CALCULUS ON MANIFOLDS

341

of half spaces in Rn . Now we can write down the general form of Stoke's Theorem, for an (n 1)-form  with compact support:

Z

M

d =

and the formula for integration by parts

Z

M

df ^  =

Z

@M

Z

@M

f



Z M

f d

R

for f 2 0 M . See Appendix E for a computer algebra version of M . Consider making a choice of inner product smoothly over the tangent spaces of (an orientable) M ; that is, choose a positive-de nite section g of T2 M = T02M . As might be expected, it induces a natural volume form g . Suppose that locally g has components gij with respect to coordinates xi ; then locally g = j det gij j 12 dx1 ^ dx2 ^


^ dxn : (On standard Euclidean space, of course, det gij = 1 in rectangular coordinates.) Such a g is called a Riemannian metric tensor for M and the fact that, at each x 2 M , g(x) is nondegenerate ensures that g is nonzero. The same formula can be used if g is a pseudo-Riemannian metric tensor; that is, we have each g(x) nondegenerate but not necessarily positive de nite, as needed for spacetime (cf. [29]). Now, g induces a dual metric tensor g which is a section of T2 M = T20M , giving an inner product on cotangent spaces. Between them, g and g then extend by tensor product to all of the other spaces (Tkm M )x , and hence by inclusion to Akx M for k-forms. Additionally, g induces a family of isomorphisms  : k M ! n k M :  7 !  called Hodge dual isomorphisms for 0  k  n. To de ne , we take any ordered base (e1 ; : : : ; en ) for Tx M and ordered bases for Akx M and Axn k M given respectively by fei1 ^ ei2 ^


^ eik j 1  i1  i2 


 ik  ng fej1 ^ ej2 ^


^ ejn k j 1  j1  j2 


 jn k  ng Then, with g(ei ; ej ) = gij and  = i1 :::ik ei1 ^


^ eik ; we have  = j det gij j 12  j1 :::jn k ej1 ^


^ ejn k where j1 :::jn k = gr1 i1 gr2 i2


grk ik i1


ik sgn(ri ) and sgn(ri ) is the sign of the permutation (i1 ; : : : ; ik ; j1 ; : : : ; jn k ) 7 ! (r1 ; : : : ; rk ; j1 ; : : : ; jn k ) : We can think of  as the `complement' or `dual' of  in the volume form g . Some special cases amplify this view:

APPENDIX C. MANIFOLDS AND BUNDLES

342

1. If 1 is the constant unit function in 0 M , then 1 = g . 2. g = ( 1) where  is the number of negative eigenvalues of g ( = 0 if g is positive de nite), Sylvester's index of g. 3. If (e1 ; : : : ; en ) is an orthonormal base for Tx M , then j det gij j = 1 and

(ei1 ^ ei2 ^


^ eik ) = (ej1 ^ ej2 ^


^ ejn k )

for any even permutation (i1 ; i2 ; : : : ; ik ; j1 ; j2 ; : : : ; jn k ) of (1; 2; : : : ; n). This simpli cation extends to k-forms by linearity. 4.  = ( 1)k(n k) 1k M . Combining the Hodge dual with the exterior derivative, we obtain the following generalizations of elementary vector calculus (recall that df = gradf for real functions): d = curl; d = div; dd =
: The familiar identities arise now as consequences of d2 = 0. The vector cross product  on R3 with the standard metric structure is related to  by

(x ^ y) = x  y : This has been a cursory look at calculus on manifolds and more information can be obtained from: Bishop and Goldberg [10], Flanders [36], Hodge [47], Singer and Thorpe [95], Spivak [99], and Warner [116]. We provide below some examples and exercises.

Ex 1. Calculations on a 2-sphere.

M = f(x; y; z ) 2 R3 j x2 + y2 + z 2 = 1g : De ne two charts (U; ) and (V; ~) for U = f(x; y; z ) 2 M j z > 0g (the `northern hemisphere') V = f(x; y; z ) 2 M j x > 0g (the `eastern hemisphere') by

 : U ! R2 : (x; y; z ) 7 ! (x; y) = (x1 ; x2 ) ; ~ : V ! R2 : (x; y; z ) 7 ! (y; z ) = (~x1 ; x~2 ) : Then on U \ V = f(x; y; z ) 2 M j x > 0; z > 0g (the NE quadrant) we have alternative coordinates: the (xi ) or the (~xi ). These determine corresponding

local basis sections:



@;@ (@1 ; @2 ) = @x @y



and





 @ @ @1 ; @~2 = @y ; @z :

~

C.3. CALCULUS ON MANIFOLDS

343

These are related (invertibly) by

@y @~1 = @2 and @~2 = @x @z @1 + @z @2

with x = (1 y2 z 2 ) 21 and y = (1 x2 z 2) 21 . Also on U \ V , we have angular coordinates: ^ : U \ V ! R2 : (x; y; z ) 7 ! (; ) = (^x1 ; x^2 ) ; where  and are the angles de ned by

9

x = cos cos  > = so = 0 on the equator and =2 at the north pole,

 = 0 in the positive x-direction, y = cos sin  > ; and  = =2 in the  y-directions. z = sin

Again we get local basis sections

^

 

@; @ @1 ; @^2 = @ @



which relate to the original choice (@1 ; @2 ) by

@y ^ @x @^1 = @x @ @1 and @2 = @ @1 + @ @2 : The local section

U \ V ! TM : (; ) 7 ! cos @^1 locally models an east-west wind on the Earth, decaying from the equator to the north pole. Similarly, U \ V ! TM : (; ) 7 ! cos @^2 models a north wind. A local ow on S2 for the vector eld

v : U \ V ! T S2 : (; ) 7 ! cos @^2 is given by

 : U \ V  ( ; ) ! S2 : (; ; t) 7 ! ((t); (t)) where the real functions ; must satisfy

d = cos ; d = 0 ; dt dt

APPENDIX C. MANIFOLDS AND BUNDLES

344

((0); (0)) = (; ) : Hence (t) =  + t cos and (t) = , and the integral curves are parts of circles of latitude. The existence and uniqueness theorem for smooth local

ows of smooth vector elds on manifolds says that through each point there is one and only one integral curve and t : x 7! (x; t) satis es t+s = t  s when t; s; (t + s) 2 ( ; ). For our ow on S2 this is satis ed for small enough s; t since t : (; ) 7!  + t cos . Moreover each t is a dieomorphism from U to t U .

2. Integral curves on Euclidean 2-space E 2 We consider M = E 2 with its standard chart. Take the vector eld

v : E 2 ! T E 2 : (x; y) 7 ! @x + x2 @y : It is convenient to use (x; y) as coordinate labels instead of (x1 ; x2 ) or (x1 ; x2 ); then we denote their induced basis elds by @x = @1 and @y = @2 . We take

the open subset of E 2 given by





U= j jx + 1j < 14 ; jyj < 41 : We seek for each a 2 U an integral curve ca : ( 12 ; 21 ) ! E 2 with ca (0) = a and c_a (t) = v(ca (t)): The dierential equation expands into (x; y) 2 E 2

c_a (t) = c_1 (t) @x + c_2 (t) @y = @x + (c1 (t))2 @y : c1 (t) = k + 1 and c2 (t) = 13 (k + t)3 + l : Taking a = (; ) as the initial point, we nd





ca (t) =  + t; 31 3 + 31 ( + t)3 : Accordingly, the local ow is given by   s : (; ) 7 !  + s; 31 3 + 31 ( + s)3 and we easily check that   t+s = t  s : (; ) 7 !  + s + t; 13 3 + 31 ( + s + t)3 :

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345

3. Components of tensors and forms on E 2

T02M

! ij dxi dxj , where ij =

 1 if i = j; 

(a) g : M ! : 7 0 if i 6= j . 2 This de nes the usual metric structure on E , that is the standard dot product on each tangent space TxM : (x1 ; x2 )

g(uk @k ; vm @m ) = ij dxi dxj (uk @k ; vm @m ) = ij uk vm dxi dxj (@k ; @m ) = ij uk vm (ki ; mj ) = ij ui vj = u1v1 + u2 v2 : (b) ! : M ! 2 M : (x1 ; x2 ) 7! dx1 ^ dx2 . This de nes the usual geometrical measure on E 2 , that is the standard parallelogram area mapped out by pairs of vectors in each tangent space TxM : !(uk @k ; vm m ) = 21 (dx1 dx2 dx2 dx1 )(uk @k ; vm @m ) = 12 (u1 v2 u2 v1 ) :

Both of (a) and (b) generalize to E n , where g has the same form but we sum now over i; j = 1; : : : ; n and ! = dx1 ^


^ dxn . Any change of chart induces corresponding changes in their local expressions. Check the role of the Jacobian!

4. Integration of forms on E 2 Let M be a closed, simply connected region in E 2 . Take v 2 1 M to be given in standard coordinates by v = 12 (x dy y dx) : Then dv = 21 (dx ^ dy dy ^ dx) = dx ^ dy ; whence

Z M

dv =

Z @M

v =)

Z M

dx ^ dy = 21

But dx ^ dy is the usual volume form for E 2 so

Z

M

Z @M

(x dy y dx) :

dx ^ dy is just the area enclosed by the curve @M:

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346

In particular, if @M is the ellipse L with

L = f(x = a cos ; y = b sin ) 2 E 2 j 0    2g then

x dy y dx = (ab cos2  + ba sin2 )d = ab d and the area of the ellipse is

Z

M

dx ^ dy = 12

Z @M

ab d = ab :

Evidently the volume form on E 3 is dx ^ dy ^ dz and its restriction to the 2-dimensional submanifold E 2 , which is the (x; y)-plane, is simply dx ^ dy. Now, the ellipse itself is a 1-dimensional submanifold of E 2 and like the circle it supports a nowhere-zero 1-form d. The standard `volume' form on the ellipse L is actually r d, where r2 = x2 + y2 , and therefore the circumference of the ellipse is

Z

L

r d =

Z 2

=a

(a2 cos2  + b2 sin2 ) 12 d

Z 2

0

0

(1 e2 sin2 ) 12 d ;

the familiar elliptic integral with e the eccentricity (1 b2 =a2 ) 21 . 5. Metric tensors on E 2 The standard Euclidean metric tensor on M = E 2 is

gx : Tx M  TxM ! R : (ui @i ; vj @j ) 7 ! u1 v1 + u2v2 and so here g = ij dxi dxj . This induces on E 2 all of the usual Euclidean geometry, including the usual volume form; it is easily generalized to E n . Another metric tensor on E 2 is given by

x : TxM  Tx M ! R : (ui @i ; vj @j ) 7 ! u1 v1 u2v2 and this is expressed as





 = ij dxi dxj where ij = 10 01 : This induces Minkowski geometry on E 2 , as used in relativity.

C.3. CALCULUS ON MANIFOLDS

347

6. Electromagnetic elds as forms The equations of the electromagnetic eld on a spacetime 4-manifold can be very neatly expressed in terms of the electromagnetic 2-form F 2 2 M . Locally, for a basis of 1-form elds (!i ),

F = Fij !i ^ !j : Taking the !i to be mutually orthogonal unit elds, the nonzero metric tensor components (gij ) are the eigenvalues of g, lying along the diagonal. The Hodge dual isomorphism gives (!i ^ !j ) = !m ^ !k where (i; j; m; k) is an even permutation of (1; 2; 3; 4). So (!1 ^ !2 ) = !3 ^ !4 , etc. Similarly, (!1 ^ !2 ^ !3 ) = !4 etc. In regions that contain negligible amounts of matter,

dF = 0 and dF = J ;

(C.1)

here J is the current density. These equations correspond to the usual vectorcalculus expression of Maxwell's equations via:

d  curl and d  divergence. Conservation of charge is expressed by divJ = 0 : This is automatically satis ed when there is negligible matter since it becomes

ddF = 0 because d2 = 0 : However, in the presence of matter, (C.1) becomes

dA = 0 ; dB = J

for some A; B 2 2M;

with A and B related by some transformation, perhaps linear. Once again, d2 = 0 ensures conservation of charge. Locally, J = Ji !i where  is the charge density. Then over a compact spacelike submanifold S of M we can measure the total charge QN and nd that

QN = = =

Z ZS ZS

!1 ^ !2 ^ !3

Z

J = d  B

@S

B :

S

APPENDIX C. MANIFOLDS AND BUNDLES

348

C.4 Bundles The tangent bundle TM to a smooth n-manifold M is present without the assumption of any further properties of M and it naturally inherits a 2n-manifold structure. Of course, TM is a special bundle of vector spaces, copies of Rn , the ber of TM . In a similar way, we can make bundles of groups with ber some xed Lie group G or more generally, bundles of manifolds. Let B be a smooth n-manifold and let F be a smooth m-manifold. A manifold E is a bundle over B with (model) ber F if

(i) there is a smooth surjection p : E ! ! B , the bundle projection; (ii) around every point x 2 B there is an open set Ux such that p Ux is dieomorphic to Ux  F . We call Ux a local triviality neighborhood. Of course, px fxg  = F for all x 2 B . We sometimes abbreviate this in the notation F ,! E ! !p B , and call E = B  F a trivial bundle.. Property (i) says that E `sits above' B in some sense. Property (ii) says that locally E looks like a product of pieces of B and copies of F . Globally, E may not be such a trivial product. For example, the Mobius strip with ber the interval [0; 1] is a twisted product of [0; 1] over S1; this is a nontrivial ber bundle over S1 with ber [0; 1]; the associated `untwisted' product is a cylinder. One of

the important achievements of algebraic topology has been to provide classi cation of the extent to which twisting occurs in a bundle, via characteristic classes and obstruction theory. Now, a bundle E over B with ber F , often just called an F -bundle over B , is particularly useful when F has not only a manifold structure but also some algebraic structure. Thus, for example, the tangent bundle TM to an n-manifold M has the structure to support linear algebra in a smoothly consistent way at each of its constituent tangent spaces. This canonical linear presence allows M to support dierentiation processes. This is achieved because sections of the projection map,

 : TM ! M : vx 7 ! x (if vx 2 Tx M ) ; have the form

s : M ! TM : x 7 ! sx (with sx 2 Tx M )

and so generalize the notion of a map

f : M ! Rn : x 7 ! f (x) from M to Rn , which is equivalent to a section f of the trivial bundle M  Rn , f : M ! M  Rn : x 7 ! (x; f (x)) : In a similar way, if G is a Lie group (that is, a manifold which is also a group having smooth group operations of composition and inversion), then we can construct a G-bundle E over a manifold B . Now, sections of the projection map of

C.5. METRICS AND CONNECTIONS

349

E onto B generalize the notion of a G-valued map on B , because the G-bundle over B actually provides (in a smooth way) one copy of G at each point of B . The precise manner in which the copies of G are smoothly arranged will depend on the particular choice of G-bundle over B . There may be many such choices, or just the trivial bundle B  G. Examples of Lie groups include Rn , the general linear groups, orthogonal groups, symplectic groups, spin groups, and discrete groups.

Ex 1. Show that SO(2) is a Lie group which is equivalent as a manifold (that is, is dieomorphic) to S1 by constructing a suitable dieomorphism. 2. Show that being dieomorphic is an equivalence relation on any set of manifolds. (The transitivity property is always extremely useful because it saves having to make direct constructions. Instead, a simpler intermediary object is used as a stepping stone; failure to exploit this has caused many examinees to waste a lot of time!) 3. Investigate the possible G-bundles over S1 when G is one of Z2  = O(1), SO(2), O(2), SL(2; R), or GL(2; R).

C.5 Metrics and connections Returning to the tangent bundle TM to a smooth n-manifold M , we recall the importance of inner products on vector spaces| these allow the de nition of lengths or norms of vectors and angles between vectors. The corresponding entity for TM is a smooth choice of inner product over its family of vector spaces fTxM j x 2 M g. Such a smooth choice is called a Riemannian metric on M . In fact, both positive de nite and inde nite quadratic forms are important in this context. The inde nite metrics on a manifold are called pseudo-Riemannian metrics and have particular signi cance in manifolds used to model relativistic spacetime (cf. [31, 30]). A manifold M equipped with a metric g is called a (pseudo-)Riemannian manifold. Given a Riemannian manifold (M; g) we1 can measure the length of any vector vx in any tangent space TxM , as (g(vx ; vx )) 2 , and the angle  between any two vectors ux; vx in the same tangent space by means of gx(ux ; vx ) cos  = : (gx (ux ; ux) gx(vx ; vx )) 12 Then it follows that the length of a curve

c : [0; 1] ! M : t 7 ! c(t) is given by

Lc(t) =

Z1 0


1

gc(t) (_c(t); c_(t)) 2 dt :

350

APPENDIX C. MANIFOLDS AND BUNDLES

Intuitively, given a metric g, we would expect to be able to deduce a sensible de nition for parallelism of vectors at a point. Pushing intuition a little more, we can also see how to de ne parallelism for vectors along a curve; that is, how to parallel transport vectors. Actually, the formal treatment of a smoothly changing inner product, and its determination of a natural de nition of smoothly changing tests for parallelism, are at the center of the contribution made by the classical innovators, Gauss and Riemann. Curvature of a manifold is de ned to be present if there is a dependence of parallel transport on the choice of path. Thus, in a standard Euclidean plane or on a cylinder, there is no curvature; they are at 2-manifolds. However, on a standard 2-sphere, the natural parallel transport of a vector, round a triangle made up of three arcs of perpendicular great circles, does not return to itself. This can be demonstrated with eect in rst courses of Dierential Geometry by parallel transporting a pencil round the head of a suitable student. The spherical triangle involved here will be seen to contain three right angles, whereas triangles in Euclidean space always have two right angles. We say that the standard sphere has positive curvature; Euclidean space has zero curvature so is called at. A saddle-shaped surface, like

S = f(x; y; z ) 2 R3 j z = x2 y2 g admits triangles with fewer than two right angles and is said to have negative curvature. See Gray [40] for a catalog of named curves and surfaces in R3 , together with Mathematica code to construct them. Plots include surfaces color-coded by various curvatures. The rami cations of the geometrical concept of curvature are very far-reaching. From the introductory comments above, it is apparent that the process of parallel transport is controlled by the rate of change of the metric along the chosen path. The local indicator of curvature is the change of the parallel transport process round a small closed path. Thus, we expect to be able to isolate the parallel transport map in terms of the gradient of the metric map and to isolate the curvature map in terms of the gradient of the parallel transport map. This is indeed the case. The parallel transport map is controlled by a linear connection; literally it connects up the parallelism structures of the individual tangent spaces. Formally, we can represent the linear connection as a smoothly varying vector-valued function on tangent vectors, that is a vector-valued 1-form, !. Then the curvature is the covariant exterior derivative (the generalized curl ) of !; that is, the curvature of a connection 1-form ! is the 2-form D! = (see below). A detailed example of the computation of connection and curvature forms and the Riemann curvature tensor is given in Dodson [29], pp. 155{158. In the middle of the last century, the concept of curvature revolutionized geometry. At the beginning of this century it did the same for cosmology, through general relativity. For the last few decades, a further abstraction of curvature, to the context of arbitrary nonmetric connections, has dominated the development of gauge theory by geometers and theoretical physicists. In 1986, we saw a Fields Medal awarded to S. K. Donaldson largely for an amazing application of gauge eld theory in algebraic topology [33]. The formal context here is that of principal bundles, and

C.5. METRICS AND CONNECTIONS

351

we shall summarize next the de nitions for connections and curvature there.

C.5.1 Principal bundles

A principal G-bundle over an n-manifold M is a G-bundle p : P ! ! M for a Lie group G such that:

(i) there is a group action P  G ! P ; (ii) the quotient of P by this action is M , giving a smooth projection P ! ! M = P=G : u 7 ! [u] : p We sometimes abbreviate this in the notation G ,! P ! ! M.

Ex 1. P = M  G is the trivial principal G-bundle. 2. P = f(x; (ei )x ) j x 2 M; (ei )x is an ordered base for Tx M g and G = GL(n; R). This is the frame bundle of M .

p A connection in a principal G-bundle G ,! P ! ! M is a smooth splitting at each u 2 P of the space Tu P into a direct sum

Tu P  = Hu P  Vu P where Vu P = ker(Tp : Tu P ! Tp(u) M ). We call Hu P the horizontal subspace at u 2 P and Vu P the vertical subspace at u 2 P . They comprise the horizontal and vertical subbundles, respectively, of TP . Now, because P is a G-bundle, it is locally trivial and so (8 x 2 M ) p (x)  = G. Next, if p(u) = x, then Tu P  = Te G = g where e is the identity in G, because B acts smoothly and vertically on p (x) and so also on Tu P . However, Te G has also the

structure of a Lie algebra induced from the commutator operation (Lie bracket). (There is a detailed discussion of Lie algebras in Hochschild [45] and of the above construction in Kobayashi and Nomizu [62].) A connection exploits this Lie algebra structure and actually appears as a Lie algebra-valued 1-form, that is, a 1-form on P taking values in the vertical subbundle V P  TP with model ber g. Just how this is done is described with examples in Dodson [29]. For our purposes, the important role of a connection is that it induces isomorphisms called horizontal lifts from tangent spaces on the base M to horizontal subspaces of the tangent spaces to P :

" : Tp(u) M ! Hu P  Tu P : v 7 ! v" : Technically, a connection splits the exact sequence of vector bundles 0 ! V P ! TP ! TM ! 0

352

APPENDIX C. MANIFOLDS AND BUNDLES

by providing a bundle morphism TM ! TP with image the bundle of horizontal subspaces. Suppose that ! is a connection in a principal G-bundle G ,! P ! ! M and it induces the bundle splitting TP  = HP  V P with projection onto horizontal components given by h! : TP ! ! HP : w = wh  wv 7 ! wh : Now, we can view the connection 1-form ! as a section of T P V P , that is as an element of 1 P V P ; so for any w 2 Tu P , !(w) = 0 2 V P if and only if wv = 0. Then d! 2 2 P V P and we de ne d!  h! = D! = to be the curvature form of the connection !. The curvature form satis es (i) d  h! = 0 ; (ii) (X; Y ) = d!(X; Y ) + 21 [!(X ); !(Y )] for all tangent vector elds X; Y on P . Note that the bracket in (ii) represents the Lie bracket (or product) in the Lie algebra g.

C.5.2 Linear connections

In the case that P = LM , the bundle of linear frames on M (cf. Cordero, Dodson and de Leon [22] for a treatise on frame bundles), the structure group is G = GL(n). L Then a connection ! in LM p! ! M is called a linear connection on M . In this case we have also another 1-form, the canonical 1-form or soldering form  : TLM ! Rn : (u; W ) 7 ! cu  TpL (u; w) ! M and cu sends where TpL is the derivative of the bundle projection pL : LM ! vectors in TpL(u) M to their components with respect to u. Hence, jV LM = 0. Now we nd two additional properties of the curvature and connection forms for a linear connection: (iii) d  h! = ^ ; (iv) (X; Y ) = d(X; Y ) + 12 (!(X )
(Y ) !(X )), where  2 2M is the torsion form of the connection !, de ned by  = d  h! : In terms of local coordinates, because V LM has ber Rn2 , the connection form ! and curvature form appear as matrices of forms on M : i dxk ^ dxl ; !ji = ijk dxk and ij = Rjkl where the kij are called the Christoel symbols. We shall see how these components can arise later (cf. equation (C.2) and preceding text). The soldering form  takes values in R and so appears as a vector of 1-forms on M : i = ji dxj :

C.5. METRICS AND CONNECTIONS

353

Then the equations (i)-(iv) above yield

d!ji = !ki ^ !jk + ij d ij = d!ki ^ !jk !ki ^ d!jk di = !ji ^ j + i of which the second and third are called Bianchi identities. p Given a principal G bundle G ,! P ! ! M , and a manifold F on which G acts, q then there is a well-de ned F -bundle F ,! (P  F )=G ! ! M called the associated F -bundle. An important example is the case when P = LM and G = GL(n). Then TM = (LM  Rn )=GL(n) is an associated Rn -bundle, and the other tensor bundles are associated F -bundles with F an appropriate tensor product of copies of Rn and (Rn ) . Every connection ! in a principal G-bundle induces a connection in every associated F -bundle F ,! E ! !q M by means of a splitting of TE into horizontal and vertical subbundles. Horizontal lifts from TM to HE are along the same lines as those from TM to HP . Along any curve c : [0; 1) ! M in M we can construct through each u0 2 p (c(0))  P a unique curve c" : [0; 1) ! P with horizontal tangent vector and p  c" = c, c" (0) = u0. The map

t : p (c(0)) ! p (c(t)) : u0 7 ! c" (t) de ned by the curve is called parallel transport along c. Parallel transport is always an isomorphism, and commutes with the action of G on P . An associated parallel transport map satis es ~t  v(c(t)) = v(c(t)). The covariant derivative of v along c is de ned to be the limit, if it exists
lim 1 ~  v(c(t + h)) v(c(t)) h!0 h h

and is usually denoted by rc_(t) v. Using integral curves c, this extends easily to rw v for any vector eld w. Evidently, the operator r is linear and a derivation:

rw (u + v) = rw u + rw v

and

rw (fv) = w(f )v + f rw v ;

it precisely measures the departure from parallelism. For the case of a connection in a frame bundle and associated bundle TM , the local appearance of r on basis elds (@i ) about x 2 M is r@i @j = kij @k where the kij are the Christoel symbols de ned earlier. For a linear connection we de ne two important tensor elds in terms of their action on tangent vector elds: the torsion tensor eld T is the section of T12 M de ned by

T (u; v) = ru v rv u [u; v]

354

APPENDIX C. MANIFOLDS AND BUNDLES

and the curvature tensor eld is the section of T13 M de ned by R(u; v)w = ru rv w rv ru w r[u;v] w : The connection is called torsion-free or symmetric when T = 0 and at when R = 0. In local coordinates with respect to base elds (@i ), T (@j ; @k ) = ( ijk ikj )@i ; R(@k ; @l )@j = (@k ilj @l ikj + hlj ikh hkj ilh )@i : Then it follows that the connection form ! 2 1 LM Rn2 of our linear connection is expressible as a matrix valued 1-form with components (C.2) !ji = ijk dxk : Hence d!ji = d( ijk ) ^ dxk = @r ijk dxr ^ dxk !hi ^ !jh = ihr hjk dxr ^ dxk and the curvature form 2 2 LM Rn2 has the local expression i dxr ^ dxk

ij = 21 Rjrk i dxr ^ dxk : = Rjrk

C.5.3 Levi-Civita connection

If M has a metric tensor eld g, then it induces a linear connection that is uniquely determined if we require that it has zero torsion and that all parallel transport maps should be isometries. These two requirements are often referred to as the symmetric and metric compatibility properties of the connection. The linear connection so determined by a metric is denoted rg ; it is called the metric or LeviCivita connection and is always used with a metric tensor unless explicitly stated otherwise (because any other choice involves extra structure). This connection determines the geodesic curves, which are those c : [0; 1) ! M having parallel tangent vector elds: rc_(t) c_(t) = 0 : There is a very important map which is de ned for each x 2 M on a neighborhood Sx of 0 2 TxM and which sends each u0 2 Sx to the point at unit distance along the geodesic through x with initial tangent vector u0 . Formally, we de ne Sx = fu0 2 TxM j 9 geodesic c : [0; 1) ! M with c(0) = x; c_(0) = u0 g :

C.5. METRICS AND CONNECTIONS

355

This is evidently non-empty and we put expx : Sx ! M : u 7 ! c(1) where c is that (unique) geodesic through c with tangent vector c_(0) = u. The uniqueness follows from the Cauchy theorem for dierential equations. The importance of the map expx , called the exponential map at x 2 M , is that every point x 2 M has a neighborhood Nx of 0 2 TxM on which expx is a dieomorphism onto its image. We say that M with its given metric structure is (geodesically) complete if expx has domain TxM for all x 2 M . We say `geodesic' completeness because this is equivalent to every inextensible (i.e., maximally extended) geodesic having domain R as a curve in M . We note that the exponential map and (geodesic) completeness can be de ned for any linear connection, metric or not.

Ex 1. Connection geometry calculations

(i) Consider E 1 with r@1 @1 = , for some constant  2 R, with respect to the standard chart. Take

c : [0; 1] ! E 1 : t 7 ! t so c_(t) = @1 : Then satis es

t : Tc(0)E 1 ! Tc(t) E 1 : 0 @1 7 ! (t)@1 d +  = 0 so (t) =  e t : 0 dt

Evidently  = 0 corresponds to the usual connection since we do not usually alter the length of vectors when we move them on E 1 . Any  6= 0 determines a non-Euclidean parallelism structure on E 1 . A similar connection could be put on S1. (ii) To nd a local expression for a parallel transport isomorphism, we consider M = E 2 with the standard chart and connection r having constant Christoel symbols 1 = 1 =1 12 21

and all other components zero.

Given the curve c : [0; 1] ! E 2 : t 7! (t; t2 ), we nd the parallel vector eld w : [0; 1] ! T E 2 : t 7 ! f (t)@1 + g(t)@2 for the two independent initial tangent vectors

w(0) = @1 and w(0) = @2 :

APPENDIX C. MANIFOLDS AND BUNDLES

356

The parallel transport condition is rc_ w = 0 and we have c_(t) = @1 +2t @2 . Substituting, _ 1 + g@ f@ _ 2 + 2tf i21@i + g i12 @i = 0 ; (f_ + 2tf + g)@1 + g_ @2 = 0 so g(t) = g(0): We solve f_ + 2tf + g = 0 for constant g to give:

case (i): f (t) = e t2 ; g(t) = g(0) = 0; case (ii): f (t) = e t2 R0t ex2 dx = k(t), say, and g(t) = 1. Then parallel transport along c is the isomorphism

  t : Tc(0)E 2 ! Tc(t)E 2 :  @1 + @2 7 !  e t2 + k(t) @1 + @2 ;

or in matrix form



 e t2 k(t)    7 ! 0 1 :

Evidently r is not compatible with the usual metric tensor on E 2 because parallel transport is not an isometry; for example,

t (@1 ) = e t2 @1 ; so r is not the Levi-Civita connection for g.

2. Calculation of geodesics in a punctured plane On an n-manifold we can always nd geodesic curves going in all direction from a point, but in general we may not be able to make them go very far. Clearly, if M has a boundary then a geodesic may not be extensible after it meets @M . Another type of inextensibility can occur if M is incomplete in some sense. Let M = E 2 n f(0; 0)g, the punctured plane with standard coordinates. Then the Euclidean connection has zero Christoel symbols and the equation of a geodesic becomes:

rc_ c_ = rc_i @i c_j @j

= ck @k + c_i c_j kij @k = 0 ;

so ck = 0 for k = 1; 2. Hence the geodesics are straight lines, as we expect for a submanifold of the Euclidean plane, but they cannot pass through the origin. Thus, for example, the geodesic

c : ( ; ) ! M : t 7 ! (2 2t; 1 t)

C.5. METRICS AND CONNECTIONS

357

which begins at (2,1) in direction 2^ ^ is only de ned for   1. Locally, the shortest distance between two points x; y on a Riemannian manifold with metric tensor g is along a geodesic from x to y. If M is incomplete then no (globally) shortest geodesic need exist. Check this for points on opposite sides of a line though the origin in E 2 n f(0; 0)g.

3. Geodesics in Schwarzschild spacetime We nd geodesics (corresponding to free particle trajectories) in Schwarzschild spacetime. Here we take M = R  (E 3 n B ) with R giving the time coordinate t and E 3 n B being the Euclidean space outside a ball of some radius k > 0 centered on the origin. We give E 3 n B the usual spherical polar coordinates (r; ; ) and view the region B as containing some spherically symmetric mass, like a star or planet. The appropriate metric tensor for this physical situation has components

21 6 (gij ) = 64

2m

0 0 0

r

1

0

2m


0 0

r

1

0 0

r2

0

0 3 0 77 0 5 r2 sin2 

for r > k > 2m, where m is the mass of the material contained in B . Denoting the coordinates (t; r; ; ) by (x0 ; x1 ; x2 ; x3 ) we nd the metric connection r has only the following nonzero Christoel symbols: 1 = 11

0 10 2 12

1 = 1 33 22 1 00 2 33 3 32



= rm2 1 2rm = 3 = 1;



1

;

13

 r2m 

sin2  ; r   = rm2 1 2rm ; = sin  cos  ; = cot  : = r 1

We recall that this connection is symmetric: kij = kji for all i; j; k. Geodesic curves satisfy rc_ c_ = 0 and we consider two cases, each with parameter s, given by g(_c; c_) = 1. (i) Circular geodesics: c(s) = (t(s); r(s); (s); (s)) with r_(s) = 0. We shall take the plane of one of these circular orbits to be (s) = =2. Denoting dierentiation with respect to parameter s by a dot, we expand rc_ c_ = 0 to give the system of equations: t = 0 =) t_ = constant,

APPENDIX C. MANIFOLDS AND BUNDLES

358

1 _ 2 + 1 t_2 00 33

= 0 =) =)





1 2rm rm2 t_2 = 0 ;

3 t_2 = rm _ 2 ;

 = 0 =) _ = constant = period 2 ; =) period = T = 2_ : From g(_c; c_) = 1 we nd

  t_2 1 2rm r2 _ 2 = 1 :

Substitution above gives circular orbits with periods

 1 T = 2r mr 3 2 ; r > 3m: In time-units, for the Sun we have 2m = 10 5 seconds and T = 1 year  107 seconds for the Earth orbit, so we deduce that the implied radius is r  500 seconds, which is what we observe. Similar data can be

checked for the Moon or other satellites orbiting the Earth. For the Earth, 2m  3  10 11 seconds. (ii) Radial geodesics: c(s) = (t(s); r(s); (s); (s)) with ;  constant. The geodesic equation reduces on  = =2 to









1 r rm2 1 2rm r_2 + rm2 1 2rm t_2 = 0 and we deduce r = m=r2 . Now r measures precisely the acceleration due to gravity at distance r from the center of a spherically symmetric mass m in agreement to rst

approximation with Newton's theory. We nd for example, in time units, that:  2m  3  10 11 sec  3 7 on the Earth r  2:1  10 2 sec r  8  10 sec;   2:5  10 13 sec r  14  10 9 sec; on the Moon 2mr  3  5:4  10 3 sec   10 5 sec on the Sun 2mr  r  18  10 5 sec:  2 sec

4. Metric connection and parallel vector elds We nd a metric connection and equations for a parallel vector eld along a given curve. Take M = (0; 2)  S1, an open cylinder with identity coordinate

C.5. METRICS AND CONNECTIONS

359

x on the interval (0; 2) and angular coordinate  on the circle S1. Consider the expression in these coordinates of the pseudo-Riemannian metric tensor (gij ) =



(1 cos x)2 0 0 (1 cos x)2



at (x; ) 2 M:

From symmetry and compatibility of the induced metric connection r with Christoel symbols ( m ij ) we have m = 1 g mk (@i gjk + @j gij ij 2

Substitution gives 1 2

@k gij ) = 21 gmk k ij ; say.





2 sin x(1 cos x) 0 0 2 sin x(1 cos x) ;   0 2 sin x(1 cos x) : = ij 2 sin x(1 cos x) 0 ij =

Hence 1

ij = 2

ij =

 

sin x 1 cos x

0 0

sin x 1 cos x

0



sin x ; 1 cos x sin x  1 cos x :

0

For the vertical-going curve

c : (0; 2) ! M : t 7 ! (t; 0) ; a parallel vector eld is

v : (0; 2) ! TM : t 7 ! f (t)@x + h(t)@ where rc_ v = 0. This dierential equation becomes the system

@x f + f @x h + h

1 11 2 12

=0 = 0;

so suitable f and h must satisfy x f and @ h = sin x h : @xf = 1 sincos x x 1 cos x

360

APPENDIX C. MANIFOLDS AND BUNDLES

5. Connection, torsion, and curvature forms (i) On E 2 with the standard coordinates, one r that is not symmetric is given by the Christoel symbols     1 = 18 ; 2 = 16 : ij ij 40 42 Its torsion form is  = 21 kij @k dxi ^ dxj = 21 (8@1 + 6@2 ) dx1 ^ dx2 + 12 (4@1 + 4@2 ) dx2 ^ dx1 = (2@1 + @2 ) dx1 ^ dx2 (with values in TxM ): So  = (2; 1)dx1 ^ dx2 (with values in R2 ). The connection form of this r is ! with
! = kij dxj @k dxi  dx1 @ 8 dx1 @    1 2 dx1 + 1 dx1 @1 6 dx1 @2 dx2 = 41 dx 2 @1 0 dx2 @2 4 dx2 @1 2 dx2 @2     so ! = 14 80 dx1 + 14 62 dx2 (with values in R22 ): (ii) The previously mentioned Schwarzschild metric tensor on M = R  E 3 n B with coordinates (t; r; ; ) is of the form 2 f 2(r) 0 0 0 3 2 6 7 (qij ) = 64 00 f 0(r) r02 00 75 with f a function of r: 0 0 0 r2 sin2  As before we let indices run 0; 1; 2; 3. Evidently, a basis of mutually orthogonal unit 1-form elds, that is of orthonormal elds, is given by
(!i ) = fdt; f 1dr; r d; r sin  d : Their exterior derivatives satisfy the structural equations d!i = !ji ^ !j where !ji = ijk !k ; i !k ^ !l : and ij = d!ji + !ki ^ !jk where ij = Rjkl Computation of the derivatives yields the following, with f_ denoting the derivative of f with respect to r: d!0 = f_ !1 ^ !0 ; d!1 = 0 ; d!2 = fr !1 ^ !2 ; d!3 = f !1 ^ !3 + cot  !2 ^ !3 :

r

r

C.5. METRICS AND CONNECTIONS

361

Then we deduce that the only non zero !ji are:

!12 = !21 = fr !2 !13 = !31 = fr !3 !10 = !01 = f_ !0 !32 = !32 = cotr  !3

_

d!12 = frf !1 ^ !2 ; _ so d!13 = frf !1 ^ !2 + fr cot  !2 ^ !3 ;   so d!10 = f_ 2 + f !1 ^ !0 ; so

so

d!23 = r12 !2 ^ !3 :

By inspection of the second structural equation we nd:





01 = f f_ + f_ 2 !1 ^ !0 ; _

02 = frf !3 ^ !0 ; _

31 = frf !1 ^ !3 + rf2 cot  !2 ^ !3 ; _

21 = frf !1 ^ !2 ; 2

3 = f 1 !2 ^ !3 ; 2

r2

03 = r !3 ^ !0 : f f_

Then from the de nition of the curvature form we obtain the components

i of the Riemann curvature tensor. For example, Rjkl 2 1 2 = R3 = f R332 223 r2 :

Einstein's equation in general relativity can be written k = R0 + R1 + R2 + R3 = 0 : Rijk ij 0 ij 1 ij 2 ij 3

It results in two dierential equations for f , reducible to

d f2 1 = 0 f 2 + r dr which admits the solution we encountered before:



f (r) = 1 2rm

 21

:

APPENDIX C. MANIFOLDS AND BUNDLES

362

C.6 Fibered manifolds As we have seen, a connection is naturally induced on a (pseudo-)Riemannian manifold and the formal ingredients yield a view of this entity as a connection in the principal bundle of linear frames. Moreover, connections can be de ned in other principal bundles, and have important applications in geometry and physics. It is then logical to ask what is the least structure that is needed to support the concept of a connection? The answer is a bered manifold, which is a submersion p:E! ! B , which means that (the derivative of) p has maximal rank everywhere (cf. [66] for more details). A connection on a bered manifold p : E ! ! B is a section of the rst jet bundle q : JE ! ! E . Now, JE consists of classes of sections of p that are equivalent up to rst derivative; an element of JE at X 2 E is represented by a linear map, the derivative T : Tp(X )B ! TX E of a section . Because  is a section, we have Tp  T = 1TB and hence T corestricts to the identity on the subbundle TB ,! TE . It turns out that JE ! !E is an ane subbundle of the vector bundle T B B TE . Locally our bered manifold connection appears as : E ! JE ,! T B B TE ; @  dxi @ ; (xi ; X ) 7 ! dxi @x i i @X 

where

@  @xi : Here, (  ) represents the class of sections of E determined by . The minus sign = i

is a convenient convention. Each connection induces the expected splitting into horizontal and vertical distributions:



TE ! HE E V E ; 
 7 ! xi ; X  ; x_ i ; x_ i i  xi ; X ; 0; X_  x_ i i :

xi ; X  ; x_ i ; X_ 

C.7 Systems of connections and universal connections The space of all connections on a given principal bundle (or bered manifold) is an in nite-dimensional vector space of sections of the appropriate jet bundle, perhaps with additional constraints to accommodate respect for group actions. Such a space is dicult to handle but contemplation of it arises naturally in cases where a family of connections must be considered. One useful way to obtain nite-dimensional bundle representations of the family of connections is to use the device of a system of connections introduced by Mangiarotti and Modugno [66] (cf. also [80]).

C.7. SYSTEMS OF CONNECTIONS AND UNIVERSAL CONNECTIONS 363 The important bonus of this approach is that on each system of connections there exists a unique universal connection of which every connection in the family is a pullback. An account of the theory for principal bundles and the frame bundle in particular, with recent applications in geometry, topology, spacetime singularities and statistical theory, can be found in Chapter 9 of Cordero, Dodson and de Leon [22]. Here we just provide the de nitions. The system of all connections on a principal G-bundle E ! ! E=G consists of the bered morphism over E :  : JE=G E=G E ! JE : (sx
G; e) 7 ! sx : Here we view JE as a subbundle of T  (E=G) TE and sx appears as the linear map (Ts)x : Tx(E=G) ! Tx E : A system of connections on a bered manifold is de ned similarly. Given

E ! !B

we want a bered morphism over E  : C B E ! JE ,! T B B TE with C ! ! B a bered manifold. We call C the system space. Then any section of C! ! B determines a unique connection on E ! ! B as a section of JE ! ! E. For the system of all linear connections on a manifold M we have two alternative views, using the tangent bundle as a bered manifold or using the frame bundle as a principal bundle. Thus we have: T : TM ! ! M with system space CT = (1T  M M TT ) 1TM  T  M M JTM ; viewing 1TM as a section of T M M TM in T M M TTM . L : LM ! ! M with system space CL = JLM=G ,! T M M TLM=G : p Let JE !j E ! ! B be the rst jet bundle over a bered manifold. Then there is a unique connection  on the bered manifold 1 : JE  E ! ! JE  having the universal property that =  for all connections 2 Sec(JE=E ). To see this, observe that the jet bundles are ane subbundles of vector bundles: JE ,! T B TE ; J (JE ) ,! T E TJE : Then de ne  : JE B E ! J (JE  E ) ,! T (JE ) T (JE  E ) ; (sx ; e) 7 ! ((X; Y; S ) 7! (X; Y; S; TsX )) :

Appendix D

Tables of Homotopy Groups The following tables collect together some information on the homotopy groups of: 1. 2. 3. 4. 5.

Spheres Sn, Special unitary groups SU (n), Symplectic groups Sp(n), Spin groups Spin(n), Stiefel manifolds Vm (Rk+m ).

365

APPENDIX D. TABLES OF HOMOTOPY GROUPS

366

D.1 Spheres In these tables, the symbol  indicates the direct sum of the groups and k Z2, for example, indicates the direct sum of k copies of Z2. The results are from Toda [111] and Mimura and Toda [77]. 

(S ) n = 2 k=1 Z k=2 Z k=3 Z k=4 Z k=5 Z k=6 Z k=7 Z k=8 Z k=9 Z k = 10 2Z

n+k

n=3

n

2 2

12 2 2 3

15 2

(S ) n=8 k=1


k=2


k=3


k=4


k=5


k=6 Z k=7 Z Z k=8 4Z k=9 5Z k = 10 2 Z  Z

n+k

120

2 2

24

(S ) k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17 k = 18 k = 19

n+k



n

(S ) k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17 k = 18 k = 19

n+k

n

2

n=2

Z  2Z

Z12 Z84

2Z

2

2

Z6 Z30 Z30 Z6 Z2 Z12 2 Z2 Z12 2 Z2

  

n=7

Z504

2



Z24 Z3 Z15 Z2 3 Z2 Z120 Z12 Z2

Z

0

2

Z6 Z24 Z4 Z120 3 Z2 4 Z2 4 Z2 Z24 Z2 Z264 Z2

 

 





n = 10

2

12

n=3  2Z 2Z

n=8

Z 0 Z Z Z Z Z Z 5Z 7Z Z  4Z Z Z Z Z Z 2

2

24

120

4

2

2

6

504

2

24

264

2

2

n=9

Z504

Z

0

2

Z6 Z16 Z4 Z240 3 Z2 4 Z2 4 Z2 Z24 Z2 Z264 Z2

 

 

Z24

Z Z

2

3Z

Z72

2



2

n=5 2Z 3Z

Z504

          

Z504

0

Z2 Z120 3 Z2 4 Z2 Z24 Z2

Z2 Z240 2 Z2 3 Z2 Z6

Z24 Z6 Z2 Z2520 Z6 Z2 Z30 2 Z6 Z2 Z24 Z12 Z4 2 Z2 Z120 Z12 5 Z2 Z132 5 Z2

   

0

Z Z2 Z60

0 0

2

2

Z6 Z30 Z30 Z6 Z2 Z12 2 Z2 Z12 2 Z2 Z132 Z2















Z2 Z2 Z24

2

n=4 5Z 6Z

Z84

2

6

6

n=7












n>k+1

2

2

2

240

























3Z Z Z

2





n = 11






















2Z Z  3Z Z Z

Z240 3 Z2 4 Z2 Z24 Z2

Z84

2

Z24 Z2 Z2 Z2 Z30 Z2 3 Z2 Z72 Z2

12

2





















2



2Z 2Z

n=6









2

n=9

n

n=5




Z Z Z



2



n=4

Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2 2 Z2 Z12 Z2

2

Z Z Z 2Z Z  2Z Z  2Z Z Z Z6 Z6 Z30

n=6

2

2

  

2

2

2

4

2

24

2

264

2

n = 10

Z504 Z12 Z6 Z16 Z2 Z240 2 Z2 Z240 Z2 3 Z2 Z24 2 Z2 Z264 Z6

    



Z504 Z4 Z240 Z6 Z12 Z2 Z60 Z6 Z504 2 Z2 4 Z2 Z24 Z6 Z2 Z1056 Z8

  

n = 11

Z504 Z2 Z6 Z2 Z16 Z2 Z240 Z2 Z2 3 Z2 Z8 Z4 Z2 Z264 3 Z2

  







D.1. SPHERES 

(S ) k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17 k = 18 k = 19

n+k



n

(S ) k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17 k = 18 k = 19

n+k

n

367 n = 12

n = 13

Z 2Z Z Z Z Z Z Z Z Z

504

2

6

2

48

4

240

Z2 4 Z2 2 Z4



Z480 Z264

n = 17













Z

Z480 2 3 Z2 5 Z2 2 Z8 Z2 Z264 2Z2

 

2

Z

2

2

2

2

264

22 23

24

25 26 27 28

29 30 31

32

33

34 35 36 37 38

39

40

41

42

(S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S ) (S 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

n+20

2

264

4

8

2

264

2

2

8

n

132

2

2

2

2

480

1

24

24

3

24

504

24

24

2

24

2

24

2

240

24

24

24

24

24

12

24

2

24

24

24

24

2

264

2

2

Z

2

2Z

2

Z Z 480

2

4Z 6Z Z 2Z Z Z  2Z 2 2

 






















Z Z Z Z  Z

 Z Z =  = 2Z  6Z =  = Z 2Z  = Z Z 2 Z =  = Z Z  =Z  Z Z =  = Z  2Z  Z  5Z =  = Z  3Z  = Z Z Z =  =Z  =Z  Z Z =  = Z Z  Z  2Z =  = Z Z  =Z  Z for n  22 )= 6

Z3 Z4 Z2 Z480 Z2 3 Z2 5 Z2 2 Z8 Z2 Z264 2 Z2

n = 20

2

2

2

                     

8



















4Z Z  2Z Z Z

2

4

2

n = 19
















2Z Z 4 Z Z Z Z Z Z 8

2

24












 

2

480

n = 16









3

8

 

n = 18

n = 15




0 Z Z Z Z Z Z Z Z 4Z 2Z Z Z Z Z

 

2

 5Z

n = 14

Z504 Z2 Z6 Z16 Z2 Z480 Z2 Z2 4 Z2 2 Z8 Z2 Z264 3 Z2

24

8

264

n>k+1 Z504

0

Z3 2 Z2



Z480 Z2 2 Z2 4 Z2 Z8 Z2 Z264 Z2

 

2

2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

368

D.2 Three special unitary and symplectic groups i=  (SU (3))  (SU (4))  (Sp(2)) i=  (SU (3))  (SU (4))  (Sp(2)) i=  (SU (3))  (SU (4))  (Sp(2))

1 2 3 0 0 Z 0 0 Z 0 0 Z 14

i i

i

4 0 0

Z2

 

Z84 Z2 Z1680 Z2 Z1680

i i

i

19

Z12 Z12

i i

Z Z

6

2Z

i

2

5

Z Z Z2

15

Z36 Z72 Z2 Z2



20

Z60 Z60

Z Z

6

3Z

2

2

2

6

7 0

Z6

0 0

8

9

Z12 Z24

Z Z

0

16

0

 

32

2



Z  Z 2Z  2Z

Z2640 Z5280

18

  

2

4

23 Z Z Z 4Z 3Z 12

2

2

2

24

2

2

D.3 Symplectic groups i (Sp(n)) inn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 0 0

Z Z2 Z2 Z2:3! Z2 Z2 Z3 Z15 Z2 2 Z2 Z12 3 Z84 2 Z2 2 Z2 Z6 Z30 Z30 Z6 Z2 Z12 2 Z2 Z12 2 Z2 Z132 Z2 2 Z2

     

2 0 0

3 0 0

4 0 0

5 0 0

n6 0 0

Z Z2 Z2

Z Z2 Z2

Z Z2 Z2

Z Z2 Z2

Z

Z

Z

Z

Z

Z Z2 Z2 Z2
7! Z2 Z2

Z Z2 Z2

Z Z2 Z2

Z Z2 Z2

Z

Z

Z

Z9! Z2 2 Z2 Z6 Z2 Z11!=Z2 Z2 Z2 Z2

Z Z2 Z2 Z2
11! Z2 Z2

Z Z2 Z2

0

0

0 0

Z5! Z2 2 Z2 Z4 Z2 Z7!=3 Z2 2 Z2 Z40 Z7!=2 Z2 Z2 Z2 3 Z2 Z32 Z2 Z44
5! 2 Z2 3 Z2



   



4n+2



(Sp(n)) =

4n+4







n  Z

(Sp(n))  =

0

0 0

3 7!

(Sp(n)) =

0 0 0

0

Z2 2 Z2 Z12 Z2 Z11!=120 Z2 2 Z2



0

0 0 0

0 Z




4n+3

0

0 0 0

2(2n+1)!

Z(2n+1)!

2

Z2

for odd n

 Z for even n 2

0 0 0

for odd n for even n

Z2

n  Z

Z60 Z60 2 Z2

Z30 Z6 Z2520 Z12 Z2 Z2520 Z2

22

Z66

Z Z Z Z

2



17

12

Z4 Z4 Z2

Z120 Z2 Z120

Z30 Z2 Z40 3 Z2 Z40

21

Z6

11

Z30

 

Z252 Z6 Z504 4 Z2 2 Z2

16

10

Z3 Z2

Z Z2 Z2

0 0 0 0 0 0 0 0 0

Z

0

13

Z4

Z6 Z4

Z

2

D.4. TWO SPIN AND TWO EXCEPTIONAL GROUPS, AND C P 2

369

D.4 Two spin and two exceptional groups, and C P2

i (G) Gni Spin(7) Spin(9) G F CP

1 0 0 0 0 0

Gni Spin(7) Spin(9) G F CP

Z2520 Z8 Z2 Z8 Z2 Z168 Z2 Z2 Z2

2

4 2

2

4 2

Gni Spin(7) Spin(9) G F CP 2

4 2

2 0 0 0 0 0

3

4 0 0 0 0 0

Z Z Z Z

0

5 0 0 0 0 0

14

   

19

20 2Z

Z2 Z2 Z6 Z2

Z504

2

Z2 Z2

Z

2

0 0

6 0 0

7

Z Z

0 0 0

Z3

0 0

15 4Z Z 3 Z 2

2

Z2 Z Z120

8 2Z 2Z

9 2Z 2Z

2

10

2

2

Z2 Z2 Z

Z2

17 2Z  2Z Z 2Z

2

8

2

6

Z8

2



0

2Z

3

Z6

Z

Z2

2

22

 



2

0 0 0

18

Z Z Z Z Z Z Z Z Z Z Z 16

8

16

8

720

3

24

 

where g = Z or 2 Z .

13

Z2 Z2

240

2

Z10395 2 Z8 4 Z2 Z11!=32 Z8 2 Z2 Z1386 Z8 Z27 or Z9 Z4 4

2

12 0 0 0 0 0

Z945 Z2835

4Z

2

21

2

8

2

Z24 Z4 Z12

Z Z2 Z Z2 Z Z2 Z Z2 Z24

0 0

Z6 Z2 Z2

16 7Z 6Z Z 2Z 2Z 3Z

11

   

Z8 Z8

2

2

23 g  5Z g  2Z g2 gZ

2 2

Z

Z 2Z 120

2

2 2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

370

D.5 Real Stiefel manifolds p = Tables for k;m k+p (Vm (Rk+m )) I. Adapted from G. F. Paechter, Quart. J. Math Oxford 7 (1956) 249{268.

p=0

p=1

p=2

p=3

m=1 m2

m=1 m=2 m3

m=1 m=2 m=3 m4 k 1 2 3 4

8s 1 8s + 3 4s + 1 8s 8s + 4 4s + 2

p=4

k 1 2 3 4 5

8s 1 8s + 3 4s + 5 4(s + 1) 8s 2 8s + 2

k=1 Z

Z

2

4s 1 4s + 1 4s 4s + 2

2

Z Z

Z

Z

Z

Z

2

Z Z

2

k = 1 2 4s 1 4s + 1 0 Z Z Z 0 2Z Z Z 0 Z 0 Z 2

2

2

2

Z 2Z Z

m=1 0

Z2 Z12 Z Z12 Z24 Z24 Z24 Z24 Z24 Z24



m=1 0 Z12 Z2 2 Z2 Z2

0 0 0 0 0 0

4s 1

2

Z2 2 Z2 Z2

Z2 Z4 Z Z2 Z2



0

4s

2

Z2

0

Z2 Z Z2 2 Z Z12 Z4 2 Z2 2 Z2 2 Z2 Z Z24 Z4 Z Z24 Z4 Z Z12



Z

4Z 3Z

2 2

Z4 Z4 2 Z2 2 Z2 Z2 Z2

4





 

 



m=4

m=5

0

2

0 2 Z Z 4Z

2

2

Z2

2

Z

m5 0 0

m=4

  

Z

2

4

Z2

Z2 Z2 Z Z12 Z4 2 Z2 2 Z2 3 Z2 Z24 Z4 Z24 Z4 Z12 Z2

m=3 2Z 0

Z

2Z

2



Z2

Z

4s + 2

Z2 2 Z2 2 Z2 2 Z2

m=3 2Z

  

 

Z





Z2 2 Z12 Z Z2 Z24 2 Z2 2 Z2 Z2 Z2 Z2 Z24 Z24 Z24



Z2 Z4 Z Z4 Z8

Z2 2 Z2 Z2 Z Z12 Z2 2 Z2 2 Z2 2 Z2 Z24 Z2 Z24 Z2 Z24 Z2

m=2

4s + 2

4s + 1

m=2



4s

Z2 Z Z2 2 Z2

2

k=1 0

Z Z

Z

5Z 2Z

Z8 Z8 Z2 3 Z2 Z2 Z2

4

Z

4

Z Z Z

2

Z Z Z

Z 2 Z2 Z2 Z2

8 8

Z

12

Z2 2 Z2 Z2 Z24 Z8 Z48 Z4 Z12

 

m6 0 0

Z

Z2

Z Z

Z

3Z

2

Z2 Z8 Z16

0

Z2

0

Z2

4

4

D.5. REAL STIEFEL MANIFOLDS p=5

m=1 m=2 0 Z Z 2Z

k 1 2 3 4 5

m=3 2Z

12

2

2

12

Z2 2 Z2 Z2

Z2 3 Z2

Z24

6

Z

Z

Z

4s + 3 8s + 1 8s + 5 4(s + 1) 8s + 6 8s + 2

0 0 0 0 0 0

0 0 0 0 0 0

1 2

0

Z2

Z8

m=4 0

m=5 0

3Z 5Z

4Z 4Z

m=6 0 2Z 3Z Z 3 Z

 or Z Z

 2 Z Z

Z

Z

2

2

2

Z

2

Z 0

2

Z

4

2Z

Z24 Z24 Z24 Z2 Z2 Z2

2

2

Z2

Z

Z

2

2

Z24

m=3

Z2 2 Z2

Z

4Z

0

p = 6 k m=1 m=2

371

Z2 Z Z8

3Z

2

4

2

Z2 Z2 2 Z2 Z4 Z4

Z2 Z2 Z2 Z8 Z8

2

Z2 2 Z Z8

or

2Z

2

4

Z2 Z2 Z Z Z8 Z Z8

 

m7 0 Z

2Z 3Z 0

2 2

Z or Z Z Z

8

Z2 Z2

4

0 0

Z8 Z16

m=4 m=5 m=6 m=7 m8 Z Z Z 2Z Z Z 2Z 3Z 2Z Z 21

2

2

2

2

II. Adapted from the thesis of C. S. Hoo, Syracuse University, 1964. Here k(r) means  k (mod r).

m = 2 pnk 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

even

odd

Z Z Z2 2 Z2 Z8 Z3 Z2 Z8 Z3

Z2 Z2 Z4 2 Z2 Z2

Z2

Z2 2 Z2 3 Z2 2 Z4 Z2 Z4 2 Z2 2 Z2 Z2









0

            

Z16 Z5 Z3 Z2 Z16 Z5 Z3 2 Z2 5 Z2 Z3 4 Z2 Z9 Z8 Z7 Z3 Z2 Z9 Z8 Z7 Z3 Z3 2 Z2 Z32 Z5 Z3 3 Z2 Z32 Z5 Z3 3 Z2 6 Z2 Z8 5 Z2 Z11 2 Z8 Z3 2 Z2

          

0

 

0 2Z Z 2Z 4Z 2Z  2Z Z 4Z 4Z 2

4

2

2

4

2

4

2

2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

372 m = 3 pnk 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0(4)

1(4)

Z 2 Z2 2 Z2 Z8 Z4 Z3 2 Z2 Z2 Z2 Z16 Z5 Z3 Z2 4 Z2 6 Z2







3

2

Z Z  Z  2Z 2Z Z Z 2Z Z Z  Z  3Z Z  4Z

Z9

8

2

4

2

3

4

2

4

2

2

4

4

2

3

2

8

2

2

2

2

32

5

3

6Z

2Z

2

2

Z4



Z3 Z2 2 Z2

5

3

     Z Z Z Z Z 2Z Z  3Z

Z4 Z2 Z16 Z5 Z3 2 Z2 3 Z2 Z4 2 Z2 4

9

3

8

7

2

3

2

2

2

 Z 2Z Z  3Z Z 6Z

Z32

 2Z  3Z Z Z  Z  5Z

Z Z Z Z 2Z Z  2Z Z  3Z Z Z  Z  2Z Z Z Z 2Z Z Z  Z  2Z

0

Z

Z8

2

7

Z

2

7

2

4

Z2

Z

9

2

2

3

2

8

2

8

2

5Z

9

5

8Z

Z8 Z11

3

2Z Z Z Z 2Z Z  Z  2Z

2

3

4

Z3

2

32

5

2

2

3

 Z 2Z Z Z Z 4Z 4

3(4)

3

2

7

4



16

4Z

Z16

2Z  Z  2Z 4

Z4

Z Z4 Z2 Z3 Z2 Z2 Z2



 



2(4)

Z2 Z2 Z Z4 3 Z2 2 Z2 Z8 Z3 Z2 2 Z2

5

4

3

4

2

8Z

2

2

Z4

 4Z

2

Z4

5Z  5Z 2

2

D.5. REAL STIEFEL MANIFOLDS m = 4 pnk 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0(4)

Z 2 Z2 2 Z2 Z Z8 Z4 Z3 3 Z2 2 Z2 Z8 Z3 Z2 Z16 Z5 Z3 Z2 4 Z2 7 Z2 Z16 Z5 2 Z4 2Z3 2 Z2 Z9 Z8 Z7 Z4 4 Z2 5 Z2 2 Z3 2 Z2 Z9 Z8 Z7 2 Z2 Z32 Z5 Z3 3 Z2 Z4 Z3 2 Z2 10 Z2 Z32 Z8 Z5 2 Z4 Z3 4 Z2 Z11 Z8 Z4 Z3 7 Z2





1(4)

Z2 Z2 Z8 2 Z2



    

                         

373

Z

0

Z2 2 Z2

2

2 2

3Z

2

4

2



2

2 2

2Z 2Z 3Z

2 2



Z Z Z Z Z Z Z 2Z Z Z Z 2Z Z Z Z 2Z 8

4

2

2

    

4

 2Z

6Z

Z8

2Z 2Z 2Z

Z8 Z2 Z16 Z5 Z3 Z2 2 Z4 2 Z4 Z16 Z5 2 Z3

2

2

3

9

2

3

7

Z2

5Z

0

Z Z

9

2Z Z4

Z2

Z2 Z4 Z3

4Z 2Z  Z Z  2Z 8

3(4)

Z Z4

Z2 Z2 2 Z2 2 Z2

8

2(4)

7

8

2

32

3

5

2

2

Z Z Z  2Z

Z8

4

8

2Z 3Z

2 2

3Z

2

2

2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

374 m = 5 pnk 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

m = 5 pnk 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0(8)

1(8)

Z 2 Z2 2 Z2 2 Z8 Z3 2 Z2 Z2 2 Z2 Z16 Z5 Z3 4 Z2 7 Z2 2 Z8 2 Z3 2 Z2 Z9 2 Z8 Z7 2 Z2 3 Z2 Z3 2 Z2 3 Z2





Z4



Z2 Z2 Z8 Z2 Z Z2

Z

Z4

2

8

4

5

3

Z2

Z Z Z Z  Z  2Z 2

8

7

4

4Z 9Z

2

5Z

2



         

2

Z Z Z

Z8

0

 

2Z  Z

9

Z2 2 Z2 Z Z8 3 Z2 3 Z2 2 Z2 2 Z2 2 Z2 4 Z2 Z16 Z8 Z5 Z4 Z3 Z2 Z8 4 Z2 5 Z2 Z3 4 Z2 Z9 Z8 Z7 3 Z2

Z2

Z3 Z2 Z8 3 Z2 Z16 Z5 Z3 2 Z4 Z4 Z2 Z3 Z9 Z8 Z7 Z2 Z8 Z4 Z8 Z3 2 Z2 4 Z2 Z32 Z5 Z3 2 Z2

2

Z4

0

Z4

2

Z16

3(8)

Z Z4

2Z 4Z

     

2(8)



        

2

2

4(8)

Z 2 Z2 2 Z2 Z16 Z4 Z3 2 Z2 Z2 Z4 Z16 Z5 Z3 4 Z2 6 Z2 Z8 Z4 Z3 2 Z2 Z16 Z9 Z7 Z4 2 Z2 3 Z2 Z3 2 Z2 3 Z2









       

5(8)

Z2 Z2 Z8 Z2 Z Z2 Z4 2 Z2 4 Z2 Z8 Z4 Z2 Z8 Z2



Z16

Z Z 5

 4Z

2

3

Z Z Z Z Z 2Z Z4

2

8

4

5Z

2

7(8)

0

Z4 Z2 Z16 Z5 Z3 2 Z4 Z4 Z2 Z3

Z2 Z2 Z Z8 3 Z2 2 Z2 2 Z2 2 Z2 2 Z2 4 Z2

Z Z Z Z Z Z Z Z 2Z 4Z Z Z  Z  3Z

Z Z Z Z Z Z Z 4Z 5Z Z 4Z Z Z Z 3Z

Z Z4

Z4



Z2

0

Z

Z2 Z8



3

7

2







9

8

7

Z2

9

Z4





6(8)

2

8

8

4

3

2

2

32

3

5

2

16

8

4

5

3

8

2

2

2

3

2

9

7

8

2

D.5. REAL STIEFEL MANIFOLDS m = 6 pnk 0 1 2 3 4 5 6 7

8 9 10 11 12 13 14 15 16 17

m = 6 pnk 0 1 2 3 4 5 6 7

8 9 10 11 12 13 14 15 16

0(8)

Z 2 Z2 2 Z2 2 Z8 Z3 Z2 Z 2 Z2 Z16 Z5 Z4 Z3 Z2 Z8 Z3 4 Z2 7 Z2 2 Z8 Z3 3 Z2 Z9 2 Z8 Z7 Z4 Z16 Z5 Z3

1(8) Z2 Z2 Z8



             Z Z Z  3Z Z Z Z  Z  4Z 3

2

4

2

9

8

375

Z2 2 Z2 Z16 2 Z2 2 Z2 2 Z2

Z

2

5Z 2Z Z 2

Z

2

2

0



Z4

Z  4Z

4

Z

2

2Z 3Z

2 2

        

Z2 Z8

0

     

Z2

8

Z2

Z3 Z2 Z Z8 4 Z2 Z16 Z5 Z3 3 Z2 Z8 Z3 2 Z4 Z3 Z3 Z9 Z8 Z7 2 Z2 Z16 Z8 Z5 Z4 Z3 Z8 Z3 4 Z2 7 Z2

2

Z4

0

Z4

4Z

Z8

3(8)

Z Z4

2 0

Z8

2(8)

2Z Z 8

Z16

2

 2Z

2

3Z

2

2

7

4

2

9Z

2

4(8)

Z 2 Z2 2 Z2 Z16 Z4 Z3 Z2 Z Z4 Z16 Z5 Z4 Z3 Z4 Z3 4 Z2 6 Z2 2 Z8 Z3 2 Z2 Z9 2 Z8 Z7 Z2 Z16 Z5 Z3



       Z Z 3

8

7

2

0

Z2 Z2 Z8 2 Z2 Z2 2 Z2

Z2

0

Z 0 Z Z Z  2Z Z Z  Z  2Z 3Z  Z 3

8

4

2

16

5

3

5Z

2

Z Z 4

2

Z8

0

Z2

Z4

7(8)

Z4

Z2 Z8 3 Z2 Z8

6(8) Z Z4

0

Z  4Z

Z8

2

Z Z  Z  4Z 4

Z2 Z2 Z8 Z2



    

Z9

5(8)

4

2

4

Z3 Z3

2

3

        

Z9 Z8 Z7 2 Z2 Z16 Z8 Z5 Z4 Z3 Z8 Z3 4 Z2

Z4

Z

2Z 4Z

2

2 2

2Z 2Z 8

Z8

2

 3Z

2

3Z

2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

376 m = 7 pnk 5 6 7

8 9 10 11

12 13 14 15 16 17

8

2

6Z 8Z

2





6Z 3Z Z Z 2

2

2Z  Z  2Z Z  2Z 8

2(8)

Z2 Z8 Z2 5 Z2

5

3

3

8

9

3

2

3

2

10 Z

2

8

5

4

4

4

2Z

Z Z Z Z Z 2Z Z 7Z

Z16

2

8

2

0

8

7

2

Z

Z8

2

Z Z 0 Z Z Z Z Z  6Z

3(8)

Z16 3 Z2 Z16 Z5 Z3 2 Z2 Z8 Z4 Z8 Z2 Z3 Z9 Z8 Z7 Z2 2 Z8 Z16 Z3 2 Z2

Z2

Z Z

Z Z4

         

2 2

Z8

 Z  2Z 3Z Z Z Z Z Z Z Z Z

Z4

3

2

2

8

4

2

16

16

2

5

3

3

4

2

3

2

2

m = 7 pnk

10 11

Z

Z Z Z Z

14 15 16 17

8 9

1(8)

2

Z16

12 13

5 6 7

0(8) 0 2Z

4(8) 0

5(8) 0

Z2

Z

Z Z Z Z 16

5Z 7Z

2 2

Z8 Z9

 Z  3Z  2Z  Z 3

2

8

Z3

5 Z2 Z8 2 Z4 Z3 Z2 Z8



0

2





0

7

Z

    

Z4 Z2 Z16 Z5 Z3 Z2 2 Z4 Z8 Z2

8

2

3

2Z

2

Z Z Z Z

Z16

8

5

4

Z3

7(8)

Z8

Z

4Z

5

4

6(8)

3

 7Z

2

Z3 Z9 Z8 Z7 Z2 Z8 Z4 Z2 Z8 Z3 3 Z2

      

Z2 Z 2 Z2 Z8

Z Z  Z  2Z 8

2

3

2

4Z 2Z  2Z 2

8

2

Z Z Z Z

Z8 Z16

4

5

2 3

D.5. REAL STIEFEL MANIFOLDS m = 8 pnk 6 7

8 9 10 11 12 13 14 15 16 17

8 9 10 11 12 13

1(8)





  

8

4

3

3

Z8

Z2

Z

6

7

8 9 10 11 12 13 14

m = 9 pnk 7

8 9 10 11 12 13

2

      

Z4 2 Z3 Z9 Z8 Z7 Z2 Z8 Z4 Z16 Z3 Z2

Z8

2

Z Z Z Z 4

8

2

2

0

2

2

4(8) 0

5(8)

16

5

4

3

6Z 7Z

2

Z8

2

Z Z  2Z  3Z Z  2Z  Z 8

4

3

9

4Z

2

4

2

8

2

0

7

0

2Z

Z8

2



   Z Z Z Z Z Z Z Z Z Z Z 9

2

4

2

0(16) 2Z Z 16

5

1(16) 3Z

3

2

Z 4 Z2 Z8 Z4 Z2 Z8 Z4 Z3

Z Z 6Z 8Z

8

4

2

2Z  Z  3Z 8

3

Z Z Z Z 0 Z Z 2Z 2Z

Z16

9

8

7

2

   

2

3

Z8

2

8

Z2

Z

4

3

4Z 5Z

2 2

2Z 2Z 8

2

Z8

2

8

Z

Z16

2

 2Z

2

0

4

2

2(16)

2

2

2

8

7

Z Z

7(8) 0 2Z

Z4 Z Z16 Z5 Z3 2 Z4 Z16 Z2 Z8 2 Z3



Z Z Z Z

2

8

6(8)

Z8 3 Z2

Z Z Z Z

          

Z16 Z5 Z3 Z2 Z4 Z2 Z8 Z3 Z2 Z9 Z8 Z7 2 Z2 Z8 Z4 Z8 Z3 Z2

3(16) Z2

Z

Z

8

4Z 4Z Z  2Z 2 2

Z Z Z Z 8

4

3

8

2

2

0

2

4(16)

  

Z16 Z5 Z4 Z3 5 Z2 6 Z2 Z8 Z3 3 Z2 Z9 2 Z8 Z7 Z2

     Z3

3Z 4Z

Z8

3

m = 9 pnk

Z8

4

2

0

2

2

2Z

2

Z

2Z

5

2

4

Z3

16

2

Z8

Z2

Z Z Z Z

5Z 2Z  Z 8

3(8)

2

5

3

Z

2(8) 2Z

Z

2

2

16

2

4Z

     0 Z 2Z Z Z Z  Z  Z 2Z Z 9Z Z  11 Z 3

Z

Z16

2

Z Z16 Z8 Z5 Z3 Z2 7 Z2 9 Z2 3 Z8 2 Z3 2 Z2 Z9 2 Z8 Z7 Z2

m = 8 pnk 6 7

0(8) 2Z

377

0

Z

2

5(16) 2Z

Z16

2

Z 3 Z 2Z Z

2

4

Z8 Z8

Z8

Z Z

2

2 3

2

Z Z 4

2

6(16)

Z Z 5

3

2 Z4 Z32 Z2 Z3 2 Z2 Z9 Z8 Z7 2 Z2 Z8 Z4 Z2 Z8 Z3 2 Z2

        

7(16)

Z Z Z  Z Z3

2

16

2

5Z 6Z

2 2

3Z Z 2Z 8

Z8

3

2

2Z 0

2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

378 m = 9 pnk 7

8 9 10 11 12 13

m = 9 pnk 7

8 9 10 11 12 13

8(16)

9(16) 3Z

Z Z Z Z

Z32

8

5

3

2

6Z 8Z

2 2

2Z  Z  3Z 8

3

Z Z Z Z 0 Z Z

Z16

9

8

7

2

   

Z 4 Z2 Z8 Z4 Z2 Z8 Z4 Z3

2

3

Z8

2

12(16)

  

Z16 Z5 Z4 Z3 5 Z2 6 Z2 Z8 Z3 3 Z2 Z9 2 Z8 Z7 Z2

     0 Z Z 3

2

10(16)

2

Z2

Z

4

13(16) 2Z Z 3 Z 2Z Z

2

Z8 Z8

Z8

Z Z

2

9

8

7

Z2

Z Z

2

8

8

Z

Z

8

4Z 4Z Z  2Z 2 2

Z Z Z Z 8

4

3

8

0

2

4

3

2 2

2

2

15(16) Z2

Z

Z Z 16

2

5Z 6Z

2

3

3

Z2

3

2

16

2

4

5

4

2

11(16)

14(16)

Z Z Z Z Z Z Z  2Z Z Z  Z  2Z Z Z Z Z Z 2Z Z16

2

4

          

Z16 Z5 Z3 Z2 Z4 Z2 Z8 Z3 Z2 Z9 Z8 Z7 2 Z2 Z8 Z4 Z8 Z3 Z2

2 2

3Z Z 2Z 8

Z8

3

2

2Z 0

2

D.5. REAL STIEFEL MANIFOLDS

379

III. k+p (Vk+m;m ) for m large and k  i (mod 8) except as otherwise noted, viz 7(16), 5(32), etc. Here Vk+m;m = Vm (Rk+m ) and \m large" means p  m 2, the stable range where k+p (Vk+m;m ) depends only on k and p. Adapted from C. S. Hoo and M. E. Mahowald, Some homotopy groups of Stiefel manifolds, Bull. Amer. Math. Soc. 71 (1965) 661{667. pni 0 1 2 3 4 5 6 7 8 9 10

0

1

Z 2 Z2 2 Z2 2 Z8 Z3 Z2



2

0(16) 2 Z  Z  Z 8(16) Z  Z 15

2

Z Z 32

15

8

5Z 7Z

2

Z8

2

 Z  2Z 3

Z4



3Z

Z16

2

2

Z8

2

Z

Z8

2

Z2

0

0

Z2 2 Z2 Z16 Z2 Z2



Z3 Z2 Z16 2 Z2

Z2 Z16 Z2

4Z

2

3

Z Z4

0

0 2Z

16

2

Z2 Z2 Z8 Z2

Z Z 15

Z2

2

Z4 Z8 Z3

Z8

2Z 2Z 3(32) 3 Z 19(64) Z  2 Z 51(128) Z  2 Z 115(128) Z  2 Z 11(16) 2 Z 3(32) Z 19(64) Z 51(128) Z 115(128) Z 11(16) Z 2 2

2

11

Z63

2Z

0

8

Z63

Z

8

4

2

8

2

16

2(32) 2 Z 18(64) Z  Z 50(128) Z  Z 114(128) Z  Z 10(16) Z 2(32) Z  Z 18(64) Z  Z 50(128) Z  Z 114(128) Z  Z 10(16) Z  Z 2

12

0

0

4

2

8

2

16

13

Z3

1(32) 17(64) 49(128) 113(128) 9(16)

Z2 Z4 Z2 Z8 Z2 Z16 Z2 Z2

  

2

2

32

3

64

3

128

3

256

3

16

3

2

2

32 64

128 256 16

Z2

APPENDIX D. TABLES OF HOMOTOPY GROUPS

380 pni 0 1 2 3 4 5 6 7

4

Z16

Z 2 Z2 2 Z2 Z4 Z2



5

Z

3

15

8

Z4

5(32) Z  2 Z 21(64) 2 Z  Z 53(64) Z  Z  Z 13(16) Z  Z 5(32) Z 21(64) Z 53(64) Z 13(16) Z

5Z

8

4(32) Z  4 Z 20(64) Z  3 Z 52(64) Z  Z  3 Z 12(16) Z  3 Z 4(32) Z  Z  Z 20(64) Z  Z  Z 52(128) Z  Z  Z 116(128) Z  Z  Z 12(16) Z  Z  Z 3

10

2

12

8

2

3

3

11 12 13

2

32

8

64

63

8

128

63

8

256

63

4

63

Z3

2

63

Z2

2

4

2

16

 2Z

2

8

2

4

4

2

2

2

2

22(32) 2 Z 14(16) Z 6(32) Z  Z 22(32) Z  Z 14(16) Z  Z 4

32

64

128

Z3

32

2

64

2

16

2

Z

2

0 7(16) 2 Z 15(16) Z 7(16) Z  Z 15(16) Z  Z 2

15

4

2

4

9

3

0

4

3Z

2

Z2 Z2 Z8 Z2

Z

Z16

2

4Z

0

0

Z 6(32) Z  Z

2Z

4

Z2

Z8 Z4

Z8

Z Z

7

Z Z4

0 0

0 0

Z16

6

Z2 Z2 Z8 Z2

32

2

16

2

3Z

2

4Z

2

16

Z2

2Z

2

Z8

Z8

2Z

Z8

2

Z8

Z

3

Z8

Z

2

Z8

0

Appendix E

Computational Algebraic Topology It's more fun to compute|Kraftwerk

No modern mathematics text is complete without some indication of how the current powerful symbolic manipulation and graphics software packages can contribute to understanding of its material. We provide some comments and examples for one of the principal systems: Mathematica [121] in the context of dierential forms. A similar package is available for Maple [19], and in the new version due out in Spring 1996 there are homological packages in the Geometry library. The only free Mathematica package that seems to be available for the linear and exterior algebraic computations typically arising in (co)homology is that of Zizza [126]. It provides functions for calculations in the exterior algebra and calculus of dierential forms on Rn . At the heart of the computational procedures is the representation of a module over a commutative ring. Here are some simple examples of the application of functions from this package; the Mathematica input and output lines are shown. Basis

yields the standard basis of forms in the speci ed variables and dimension.

Basis[{x,y,z,u,v},2] {(1) dx^dy, (1) dx^dz, (-1) du^dx, (-1) dv^dx, (1) dy^dz, (-1) du^dy, (-1) dv^dy, (-1) du^dz, (-1) dv^dz, (1) du^dv}

speci es the coordinate set for a chain and Boundary derives its boundary chain; here's a 2-ball and its boundary circle

Chain

a=Chain[ {x -> r Cos[theta], y -> r Sin[theta]},{r, 0, R},

381

APPENDIX E. COMPUTATIONAL ALGEBRAIC TOPOLOGY

382

{theta, 0, 2Pi}] Boundary[a] Chain[1, {x -> R Cos[theta], y -> R Sin[theta]}, {theta, 0, 2 Pi}]

We can integrate the form d[x,y]=(1) ary as follows

dx^dy over the

chain a, and over its bound-

Integral[d[x,y],a] 2 Pi R Integral[d[theta],Boundary[a]] 2 Pi

Tensor products are generated using TensorProduct[t[x], t[x]] + TensorProduct[t[y], t[y]] (1) d[x] o d[x] + (1) d[y] o d[y]

Exterior dierentiation is given by d d[3x^2y^6] 2 5 6 (18 x y ) dy + (6 x y ) dx

Co-exterior derivatives are found using cod with a speci ed metric; the standard metric in R2 is t[x,x] + t[y,y]= (1) d[x] o d[x] + (1) d[y] o d[y] cod[ 3x^2y^6 d[x] ^^^ d[y], t[x,x] + t[y,y]] 2 5 6 (18 x y ) dx + (-6 x y ) dy

The Hodge dual operation is HodgeStar[3x^2y^6 d[x] ^^^ d[y], t[x,x] + t[y,y]] 2 6 3 x y

We nd a coboundary of the closed form r

d[r, theta] in

polar coordinates using

HomotopyOperator[ r d[r, theta], Coordinates -> {r, theta}] 2 r -(r theta) (--) dtheta + (----------) dr 3 3

383 Using the de nition cod[ d[x], metric ed metric, the Laplace operator is

] + d[ cod[ x, metric] ]

for a speci-

Laplace[3x^2y^6, t[x,x] + t[y,y]] 2 4 6 -90 x y - 6 y

The operator Orientation gives a choice of unit vector of the top-dimensional exterior algebra on the variables in a speci ed metric; it bases its choice on the Sort function. Orientation[t[r,r] + r^2 t[theta, theta] + t[z,z]] 1 (---------) dr^dtheta^dz -2 Sqrt[r ]

Under a transformation of coordinates, we can pullback a tensor Pullback[ d[x,y], {x -> r Cos[theta], y -> r Sin[theta]}] 2 2 (r Cos[theta] + r Sin[theta] ) dr^dtheta

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Index Boldface page numbers indicate that the item occurs on that page in boldface, Italic page numbers indicate occurrence in Italics. Terms being de ned are usually in boldface in the text.

attaching and removing cells, 73 Axiom of Choice, 6, 296

abelian group, 301 monoid, 226 abelianization, 156 absolute homotopy group, 58 abstract simplicial complex, 144 action eective, 305 free, 305 properly discontinuous, 64 transitive, 305 acyclic, 128 map, 234 space, 234 adjointness of S and , 44 antipodal, 48 degree theorem, 119 maps, 48 theorem, 48 antisymmetric relation, 295 antisymmetry, 295 arrows, 297 associated bundle, 215, 353 principal bundle minimal, 225 associative, 297 Atiyah-Hirzebruch spectral sequence, 195 theorem, 195 atlas, 331

back

k-face of singular cube, 145 q-face of singular simplex, 150

ball extension, 25 barycenter, 81, 90, 154 barycentric coordinates, 81 subdivision, 83, 154 base of topology, 311 point, 21 space, 62, 212 basis elds, 338 Betti number, 117, 157 Bianchi identities, 353 Bieberbach group, 307 bigraded chain complex, 194 ltration, 190 bijectivity, 296 binary operation, 301 Bockstein, 111, 256, 289, 292 Bolzano-Weierstrass theorem, 318 bordism, 247 Borel theorem, 249 Borsuk, 25 -Ulam theorem, 121 antipodal theorem, 120 boundary, 311 393

394 of a manifold, 340 of a simplex, 83 operator, 126, 130, 148, 251 bounded below, 193 braid group, 204 Brouwer xed point theorem, 48, 118 Bun, 226 bundle, 29, 348 associated, 215 classifying functor, 224 cotangent, 334 ber of, 348 of coecients, 217 projection, 348 tangent, 334 vector, 215 B Z2 and E Z2, table, 223 canonical 1-form, 352 cap product, 169 category, 297, 324 of bers, 232 Cayley numbers, 66 C -chain cohomology with coecients, 130 complex, 126 map, 126 C ech theory, 172{174, 210 cell, 69 attaching, 73 complex, 69 in a spectrum, 184 removing, 73 cellular approximation theorem, 76, 92 hull, 70 map, 70 chain complex, 125{140 bigraded, 194 singular, 148 contractible, 128 homotopy, 127 C hain, 127 change-of-coordinates formulas, 216 characteristic

INDEX classes, 348 Chern, 250 of a bundle, 248 of a bration, 278 Pontrjagin, 250 Stiefel-Whitney, 249 universal, 248 map, 70 chart, 331 Chern classes, 250 Christoel symbols, 352 CIA theorem, 207 classifying map, 28, 221 space, 28, 272 of G, 221 Cliord-Klein theorem, 307 clopen, 319 close simplicial approximations, 154 closed, 311 m-forms, 131 closure, 311 - nite, 71 co- (pre x), 313 cobordism, 247 ring, 247 coboundary operator, 130 cochain alternated, 251 complex, 130 fundamental, 251 coclassifying map, 33 cocycles, 131 C ech, 210 coecient group, 108, 272 coequalizer, 300 co ber, 35 co bration, 21, 33 composition, 35 property, 33 co nal subspectrum, 183 co nite-open topology, 324 cofunctor, 10, 298 cohomologous, 210 cohomology

INDEX classes, 108, 253 cobordism, 247 continuous, 173 nonabelian, 210{219, 250{256 H-group, 170 operations, 172 products, 150{151, 168{170 reduced spectral, 188 spectral sequence, 200 theory, 107, 108 reduced, 108 unreduced, 108 cohomotopy stable, 189 cokernel, 302 colimit, 180 commutative group, 301 commutator, 338 subgroup, 156 commuting up to homotopy, 37 compact, 8, 316 -open topology, 39 compactly generated, 21 complement, 233 complementary degree, 190 complete, 324

ow, 338 linear connection, 355 completely regular space, 328 compose, 297 composite morphism of spectra, 184 cone, 22 con guration space, 200 connected, 8, 319 connecting morphism, 59, 110, 131 connection, 351

at, 354 geometry calculations, 355 symmetric, 354 torsion-free, 354 continuous, 311 contractible, 19, 23, 37, 45

395 space, 19, 21 converges spectral sequence, 195 conversion of lifting to extension, 31 convex linear structure, 80 coproduct, 170 coretraction, 54 cotangent bundle, 334 covariant derivative, 353 covering, 62 projection, 62 space, 62 n-connected, 264 regular, 200 crystallographic group, 307 cubical singular chain complex, 145 chains, 145 homology theory, 146 cup product, 150, 168 curl, 350 curvature, 354 form, 352 negative, 350 positive, 350 zero, 350 curve, 321, 334 cut point, 157 CWcomplex, 70 inclusion co bration, 73 decomposition, 36 homology, 135{142 cycles, 131 cyclic group, 304 cylinder triangulation, 84 de Rham cohomology, 166 cohomology theory, 166 theorem, 168 deformation retract, 23, 52, 109 weak, 109 degenerate singular cube, 145 degree

INDEX

396 complementary, 190 ltered, 190 in a ltration, 190 of a map, 118 theorem, 118 total, 190 dense set, 311 derivation, 337 derived complex, 154 dieomorphic, 334 dieomorphism, 334 dierentiable, 333, 334 dierential, 130 forms, 165 manifolds (smooth), 165 dimension axiom, 108 of a cell, 70 of a simplex, 152 of a simplicial complex, 152 direct limit, 180 product of groups, 303 disconnected, 319 disk extension, 23 divisible group, 133 domain, 295 dual numbers, 233 duality, 36 edge

group, 87 loop, 86 eective action, 305 Eilenberg-Maclane space, 261 electromagnetic 2-form, 347 charge density, 347 current density, 347 elds as forms, 347 E (n), 306 entire relation, 295 epic, 300 epimorphism, 30, 300 equalizer, 300 equivalence, 295

equivalent orientations, 151 etale map, 176 space, 176 Euclidean group, 306 Euler -Poincare theorem, 162 characteristic, 84, 117 class, 243 exact, 54 at, 54 homology sequence of a pair, 107 of a triple, 110 homotopy sequence of a bration, 67 of a pair, 59 of a triple, 68 m-forms, 131 sequence, 54 long, 110 short, 54 split, 54 excision (co)homology, 106{108 homotopy, 76, 96 excisive pair, 158 triad, 114 exponential map, 45, 137 of a linear connection, 355 Ext, 133{134 extend, 5 extension functor, 133 of a map, 5 of bundle structure group, 214 of one group by another, 134 problem, 5 to join, 26 exterior derivative, 339 dierentiation, 335 E Z2 and B Z2, table, 223 face, 82

INDEX of a simplex, 152 operator, 145 FHE, 236 ber, 62, 212, 348 bundle, 62 bundle with model ber, 176 dimension, 227 homotopic, 236 homotopy, 236 homotopy equivalence, 236 model, 29 over, 219 standard, 28, 62 type, 29 bered coproduct, 299 manifold, 362 product, 226, 299 space, 29 bration, 21, 27 bundles, 67 homotopy type of bers, 236 eld, 300 lterbase limit, 9 ltered degree, 190 ltration of a space, 190 of homology, 190 nite -closed topology, 324 -complement topology, 324 homology, 117 intersection property, 317 type, 171 FIP, 317 rst order obstruction, 31, 36 ve lemma, 55

at, 350 connection, 354 manifold, 350

ow, 124 complete, 338 global, 338 local, 338 form

397 closed, 339 exact, 339 locally exact, 339 vector-valued, 339 volume, 339 four lemma, 55 frame bundle, 213, 351 free action, 305 chain complex, 128 group, 303 homotopy, 61 product of groups, 303 resolution, 133 Freudenthal suspension theorem, 76, front

98

k-face of singular cube, 145 p-face of singular simplex, 150

frontier, 311 functor, 10, 298 fundamental class, 278 group, 45 in action, 46 nonabelian, 46 of a product, 47 of a wedge, 46 Theorem of Algebra, 47

G-

bundle, 212 classi cation theorem, 213 space, 214 left, 215 torsor, 212 general linear group, 301 geodesics circular, 357 curves, 354 punctured plane, 356 Schwarzschild spacetime, 357 geometric simplicial cohomology, 167 simplicial complex, 82, 151 geometry

398 Euclidean, 346 Minkowski, 346 germs, 178 Geroch gh theorem, 288 spin theorem, 286 ghost, 95, 260 gift wrap theorem, 121 GL(n), 301, 307 global

ow, 338 gluing lemma, 77 graded C -object, 130 group, 108 Grassmann algebra, 168 manifold, 100 product, 169 group, 300, 301 abelian, 301 action, 305 properly discontinuous, 64 symmetry of, 46 commutative, 301 completion, 227 homomorphism, 301 homotopy, 44 perfect, 234 quotient, 302 Gysin sequence, 243

H-

cogroup, 37, 40, 41 cogroup functor, 43 group, 37, 38 group cofunctor, 40 isomorphisms, 44 space, 38 HahnBanach theorem, 7 Mazurkiewicz theorem, 322 hairy ball theorem, 49, 121 ham sandwich theorem, 122 Hausdor, 8 space, 35, 315

INDEX height of generator, 171 Heine-Borel theorem, 318 Hirsch theorem, 288 Hodge dual isomorphisms, 341 Hom, 129 properties, 129{130 homeomorphic, 23, 311 homeomorphism, 311 homogeneous space, 64, 306 homological algebra, 234 homologous cycles, 131 homology, 194 bordism, 247 chain, 127 classes, 108 cooperations, 172 of spheres, 113, 115, 159 Pontrjagin product, 171 reduced singular, 149 spectral, 188 simplicial singular, 148 theory, 106, 107 reduced, 108 unreduced, 108 homomorphism, 300 homotopic, 17, 185 constant maps, 27 maps of pairs, 57 homotopy, 17, 185 associative, 38, 41 classes of maps, 21 cofunctor, 271 commutative, 38, 41 equivalence, 29 equivalent spaces, 18, 21 excision theorem, 76, 96 free, 61 group, 37, 44 of a wedge, 60 of Lie groups, 102 relative, 57 identity, 38, 41 inverse, 19, 38, 41

INDEX killing, 262 lifting property, 27 relative, 17 stable, 189 type, 18, 21 Hopf, 37 algebra, 171 degree theorem, 120, 266 H-space theorem, 171 sphere bundles, 65, 99, 100 trace theorem, 162 horizontal lift, 351 subbundle, 351 subspace, 351 Hurewicz bration, 28 isomorphism theorem, 141 proof, 197 identi cation space, 35 topology, 313 identity element of a group, 301 morphism, 297 image, 295 inclusion, 313 co bration, 35 opposite a vertex, 150 index of subgroup in group, 302 inessential, 17, 23, 32 i n-trivial, 275 inf topology, 313 1-equivalence, 93 injectivity, 296 inner product space, 230 integral curve, 338 interior, 79, 311 internal product, 150 invariance of domain, 113 inverse arrow, 300 limit, 180, 200 isic, 300

399 isomorphic, 301 isomorphism, 300, 301 isotropy subgroup, 305 iterated loop space, 40 join, 25, 222 n-fold, 222

K-theory, 228 Kan action 2-dimensional, 254 principal, 254 kernel, 53, 301 killing homotopy, 261 Kirby-Siebenmann theorem, 288 Klein bottle, xi, 72, 88, 125, 140, 284 as K (; n), 266 as orbit space, 50, 306 homology, 140 punctured, 19, 47 triangulation, 84 K (; n), 261 Kronecker index, 132 product, 132, 278 Kunneth formula, 135 Lebesgue lemma, 318 number, 318 Lefschetz xed point theorem, 123, 163 number, 122 left coset, 302 G-space, 215 limit, 180, 299 Leibniz class, 213 cocycle, 213 Leray -Hirsch theorem, 245 -Serre spectral sequence, 239 theorem, 239 theorem, 211

INDEX

400 Levi-Civita connection, 354 Lie algebra, 351 bracket, 338 group, 305, 348 lift, 6 lifting of a map, 6 of bundle structure group, 214 problem, 6 through Postnikov towers, 279 limit, 180 in category, 298 of a diagram, 180 point, 311 line (element) eld, 290 bundle, 30 linear, 214 connection, 350, 352 map of a simplicial complex, 82 space, 300 linearity, 7 local coecients, 217

ow, 338 homeomorphism, 176 triviality neighborhood, 348 locally a product, 176 contractible, 30 nite, 318 trivial, 29 loop space, 38, 39 Lorentz group, 66 Lusternik-Schnirelmann theorem, 122 Lusztig's theorem, 230

m-

boundaries, 131 coboundaries, 131 cocycles, 131 cycles, 131 form, 335 simplex, 144

manifold smooth, 331 topological, 332 with boundary, 340 map, 295 acyclic, 234 admissible, 236 mapping cone, 22 cylinder, 7, 22, 59 bration, 28 spectrum, 187 path space, 27 Marcus's theorem, 291 Mayer-Vietoris sequence, 114, 158 relative simplicial, 158 singular, 159 meteorology theorem, 121 metric compatible connection, 354 connection, 354 and parallel vector elds, 358 tensor, 341 Milnor construction of classifying space, 222 minimal associated principal bundle, 225 triangulation, 84 ( 1)-skeleton, 82 model ber, 29, 219 module, 300 projective, 227 Mobius strip, 1 triangulation, 84 moles, 75, 93 monic, 300 monoid abelian, 226 monomorphism, 36, 300 Moore -Postnikov decomposition, 31, 267 space, 261 more is better, 170 morphism, 297

INDEX connecting, 59, 110, 131 of sheaves, 177 of spectra, 183 M (; n), 261

n-

connected, 44, 91 pair, 61 dimensional obstruction set, 276 equivalence, 93 manifold, 331 skeleton, 70 trivial map, 275 natural equivalence, 298 isomorphism, 298 transformation, 298 negative curvature, 350 normal, 318 space, 315 subgroup, 302 nullhomotopic, 17, 22 i n-trivial, 275

objects, 297 obstruction, 249 rst, 259 rst order, 31, 36 xed-point free, 123 nth , 259 primary, 31, 36 principal co bration, 35 principal bration, 31 theory, 348 to vacuity, 149 octonions, 66 O(n), 301 one-point compacti cation, 329 open, 311 cover, 316 sets, 310 orbit, 305 space, 305 order of group, 302

401 partial, 312 ordinary (co)homology theory, 108 all agree, 196 orientable ber bundle, 217 bration, 235 GL(n)-bundle, 214 orientation, 216 bundle, 214 of a simplex, 151 oriented simplices, 151 origami, 82, 84 orthogonal group, 301 orthonormal, 360 pair, 56 paracompact, 21, 319 parallel transport, 350, 353 parallelizable, 228, 334 partial order, 295 for a topology, 176 partition, 313 of unity, 340 path -connected, 320 lifting property, 29 phantom, 95, 260 Poincare duality, 169 pointed spaces, 21 polyhedron, 82, 152 Pontrjagin classes, 250 product, 171 poset, 312 positive curvature, 350 Postnikov decomposition, 262 invariant, 279 section, 264 system, 262 tower, 262 preserve diagrams, 10 presheaf, 176 primary obstruction, 31, 36 principal

INDEX

402 bundle, 212, 351 co bration, 33 bration, 27 bration property, 28 problems, 10 product space, 30 projective space, 21, 50, 67, 88, 103, 139, 151, 266, 290, 307, 318, 327, 329, 369 and xed points, 125 and orientation, 289 as classifying space, 223 as CW-complex, 71, 139 as Grassmannian, 101 as K (; n), 261 as M (; n), 261 as orbit space, 306 base of bration, 65 cohomology, 243, 247 essential map, 260 homology, 140 homotopy, 68 in nite, 71 triangulation, 84 prolongation, 214 properly discontinuous, 64 group action, 64 pseudoRiemannian manifold, 349 Riemannian metric, 341, 349 simplicial complex, 86 triangulation, 86 pullback, 299 bundle, 219 punctured Klein bottle, 19, 47 plane, 356 torus, 19, 47 pushout, 299 quantum eld theory, 218 quaternions, 65, 167, 306, 318 quotient topology, 313

r-dimensional, 82 realization, 82, 152

reduced (co)homology theory, 108 chain complex, 149 homology, 149 homology theory, 197 pointed space construction, 23 spectral cohomology cofunctor, 188 homology functor, 188 reduction of bundle structure group, 214,

220

theorem, 116 re nement, 318 re exive relation, 295 re exivity, 295 regular cover, 200 space, 315 rel, 17 relation, 295 relative cubical singular homology, 147 CW-complex, 71 homotopy, 17 homotopy functor, 57 homotopy set, 57 Mayer-Vietoris sequence, 158 removing and attaching cells, 73 representable functor, 221 representation, 214 restriction functor, 106 map in a presheaf, 177 retract, 35, 109 deformation, 23, 52, 109 weak, 109 retraction, 8, 23, 25, 46, 99 problem, 8 reverse arrows, 10, 298 Riemannian manifold, 349 metric, 341, 349 existence, 220

INDEX right coset, 302 inverse, 54 limit, 180, 299 ring, 200, 300

s-equivalent, 230 SAT, 89, 154 section, 54, 218, 296 of a presheaf, 177 semidirect product, 306 separation properties, 8, 315 Serre bration, 28 sequence, 244 ses, 54 Set , 53 SFHE, 236 sheaf, 177 space, 176 shea cation, 179 short exact sequence, 54 simple covering, 211 simplices, 144 simplicial approximation, 153 approximation theorem, 89 complex, 46, 82 homology, 153 map, 83, 152 pair, 152 pair map, 152 simply connected, 45 singleton, 19 space, 21 singular chain complex, 148 m-cube, 145 m-simplex, 148 Mayer-Vietoris sequence, 159 skeletal, 232 skeleton, 82, 152 SL(n), 307 smash product, 41 smooth, 331

403 manifold, 331 triangulation, 165 soldering form, 352 SO(n), 301 source, 180 space acyclic, 234 completely regular, 328 Hausdor, 315 normal, 315 regular, 315 second countable, 326 separable, 326 system, 363 Tk , 315 spans, 81 spectral sequence, 194 spectrum, 182 on a space, 182 sphere as CW-complex, 71 as M (; n), 261 as K (; n), 261 holds water, 98 triangulation, 84 spin group, 306 structure, 287 Spin(n), 306 spinor group, 66 structure, 287 split, 54, 230, 351 Sp(n), 306 stabilization, 224 stabilizer, 305 stable, 102 cohomotopy theory, 189 equivalence, 227 homotopy theory, 189 stably equivalent, 227 trivial, 227 stalk, 175, 178 standard

INDEX

404 interval homeomorphism, 18

m-simplex, 147 n-cell, 70 n-simplex, 80

star, 152 Steenrod class, 217 map, 217 stellar subdivision relative, 154 Stiefel -Whitney classes, 249 theorem, 287 variety, 100 Stokes's theorem, 168 subbase, 311 bundle, 220 category, 298 complex, 70, 152 cover, 316 group, 301 simplex, 82 space topology, 313 summation convention, 337 SU (n), 306 sup topology, 313 surjectivity, 296 suspension, 38, 41, 245 and loop exact sequence, 56 functor, 106 symmetric connection, 354 group, 303, 307 relation, 295 symmetry, 295 symplectic groups, 306 system of all connections, 363 of coecients, 250 of connections, 362 tangent bundle, 334 space, 332 vector, 334

eld, 49

target, 180 tensors, 337 Thom class, 247 isomorphism theorem, 247 Tietze theorem, 9, 316, 318 Tk -space, 315 Top, 21 Top , 21 g, 40 Top topological group, 305 manifold, 332 space, 310 topology, 310 co nite, 322 coinduced, 313 greatest lower bounding, 312 induced, 313 inverse image, 313 largest, 313 least upper bounding, 312 quotient, 313 smallest, 313 weak, 71 Topp , 56 Tor, 134{135 torsion, 353 -free connection, 354 form, 352 functor, 134 torsor, 212 torus, 1, 23, 88, 125, 284, 332, 334 as CW-complex, 72, 138 as K (; n), 266 as orbit space, 50, 306 as quotient space, 19 fundamental group, 45 homotopy, 19 punctured, 19, 47 solid, 21 triangulation, 84 with graph, 3 total

INDEX degree, 190 order, 295 space, 62, 212 transcription, 9 transgression cohomology, 245 homology, 244 transition functions, 212 transitive action, 305 relation, 295 transitivity, 295 triangular diagrams, 6, 297{298 triangulation, 82 minimal, 84 smooth, 165 triple, 68 trivial bundle, 30, 348 principal bundle, 351 twisted bundle, 30 coecients, 217 Tychono theorem, 318

U (n), 306

under, 33 uniqueness of image, 296 unitary group, 306 universal bundle, 221 characteristic classes, 248 coecient theorem, 133, 134 connection, 363 element, 272 unreduced (co)homology theory, 108 unreduction theorem, 116 Urysohn lemma, 9, 316, 318 Van Kampen theorem, 88 vector bundle, 215 space, 300 vertical subbundle, 351

405 subspace, 351 vertices, 81, 144 Wang sequence, 244 weak deformation retract, 109 bration, 28, 67 homotopy equivalence, 93 retract, 109 topology, 71 wedge axiom, 136 product, 19, 40 WHE, 116 Whitehead theorem, 76, 95 Whitney embedding theorem, 333 sum, 226 Witt ring, 230 words in free group or product, 303 worms, 75, 93 Zeeman comparison theorem, 248 zero curvature, 350