Introduction to Piecewise-Linear Topology [1 ed.] 9783540111023, 9783642817359

Citation preview

STUD~

~l'6

~tJbER

C. P. Rourke B. J. Sanderson

Introduction to Piecewise-Linear Topology

With 58 Figures

Springer-Verlag Berlin Heidelberg New York 1982

Colin P. Rourke . Brian J. Sanderson The University of Warwick, England

Revised printing of Ergebnisse der Mathematik' und ihrer Grenzgebiete, Vol. 69,1972

AMS Subject Classifications (1970): Primary 57 A XX, 57 C XX, 57 D XX Secondary 50 XX, 52 XX, 53 XX, 54 XX, 55 XX, 57 XX, 58 XX ISBN-13 :978-3-540-11102-3 e- ISBN-13 :978-3-642-81735-9 001: 10.1007/978-3-642-81735-9 Library of Congress Cataloging in Publication Data Rourke, C. P. (Colin Patrick), 1943Introduction to piecewise-linear topology. "Springer study edition". Bibliography: p. Includes index. , 1. Piecewise linear topology. 2. Manifolds (Mathematics). 3. Differential topology. I. Sanderson, B. J. (Brian Joseph), 1939-. II. Title. QA613.4.R68 1982 514'.2 81-18311 ISBN-13:978-3-540-11102-3 (U.S.: pbk.) AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to Verwertungsgesellschaft Wort, Munich. @) by Springer-Verlag Berlin Heidelberg 1972, 1982

214113140-543210

Preface

The first five chapters of this book form an introductory course in piecewise-linear topology in which no assumptions are made other than basic topological notions. This course would be suitable as a second course in topology with a geometric flavour, to follow a first course in point-set topology, andi)erhaps to be given as a final year undergraduate course. The whole book gives an account of handle theory in a piecewiselinear setting and could be the basis of a first year postgraduate lecture or reading course. Some results from algebraic topology are needed for handle theory and these are collected in an appendix. In a second appendix are listed the properties of Whitehead torsion which are used in the s-cobordism theorem. These appendices should enable a reader with only basic knowledge to complete the book. The book is also intended to form an introduction to modern geometric topology as a research subject, a bibliography of research papers being included. We have omitted acknowledgements and references from the main text and have collected these in a set of "historical notes" to be found after the appendices. Note on the Paperback Edition

For the paperback edition, the main text has been revised only where necessary in ord.er to correct mistakes in the original edition and we are very grateful to readers of the original edition for pointing out these mistakes. The bibliography has been revised to take account of publications, relevant to the topics listed, which have appeared since the original edition. No attempt however has been made to cover areas which have developed in the meantime (for example the recent work of Thurston et al. on geometric structures).

Table of Contents

Chapter 1. Polyhedra and P.L. Maps Basic Notation . joins and Cones. . . Polyhedra. . . . . . Piecewise-Linear Maps The Standard Mistake P.L. Embeddings Manifolds . . . . . Balls and Spheres . . The Poincare Conjecture and the h-Cobordism Theorem Chapter 2. Complexes . Simplexes. . . Cells. . . . . Cell Complexes Subdivisions . Si,~plca Complexes. Simplicial Maps . . . Triangulations. . . . Subdividing Diagrams of Maps Derived Subdivisions. . . . . Abstract Isomorphism of Cell Complexes Pseudo-Radial Projection. External Joins. . . . . . . . . . . . Collars. . . . . . . . . . . . . . . Appendix to Chapter 2. On Convex Cells Chapter 3. Regular Neighbourhoods Full Subcomplexes. . . . Derived Neighbourhoods . Regular Neighbourhoods .

. . . .

1 1 1 2 5 6 7 7 8 8 11 11 13 14 15 16 16 17 18 20 20 20 22 24 27 31 31 32 33

Table of Contents

Regular Neighbourhoods in Manifolds . . . . . Isotopy Uniqueness of Regular Neighbourhoods . Collapsing . . . . . . . . . . . . Remarks on Simple Homotopy Type . Shelling . . . . Orientation . . . . . Connected Sums. . . SchOnflies Conjecture.

VII

34 37 39 39 40 43 46 47

Chapter 4. Pairs of Polyhedra and Isotopies Links and Stars . . . . . Collars . . . . . . . . . . . . . . . ..... . Regular Neighbourhoods Simplicial Neighbourhood Theorem for Pairs Collapsing and Shelling for Pairs Application to Cellular Moves. Disc Theorem for Pairs. . . . ~ Isotopy Ex~nsio . . . . . . .

50 50

Chapter 5. General Position and Applications. General Position. . . . . . Embedding and Unknotting . . . . . . Piping . . . . . . . . . . . . . . . Whitney Lemma and Unlinking Spheres Non-Simply-Connected Whitney Lemma

60 60 63

Chapter 6. Handle Theory . Handles on a: Cobordism . Reordering Handles . . . Handles of Adjacent Index Complementary Handles Adding Handles . . . . . Handle Decompositions. . The CW Complex Associated with a Decomposition The Duality Theorems . . . . . . Simplifying Handle Decompositions Proof of the h-Cobordism Theorem The Relative Case . . . . . . . The Non-Simply-Connected Case Constructing h-Cobordisms . . .

74 75

Chapter 7. Applications . . . . . . Unknotting Balls and Spheres in Codimension ;:;;; 3 . A Criterion for Unknotting in Codimension 2. . .

91 91

52 52 53 54 54 56 56

67 68 72

76 76 78 80 81 83 84 84 87 87

88 90

92

vm

Table of Contents

Weak 5-Dimensional Theorems Engulfing . . . . . . . . . Embedding Manifolds . . . Appendix A. Algebraic Results . A. 1 Homology . . . . . A. 2 Geometric Interpretation of Homology A. 3 Homology Groups of Spheres A. 4 Cohomology. . . . A. 5 Coefficients'. . . . A. 6 Homotopy Groups . A. 7 CW Complexes . . A.8 The Universal Cover Ap~ndix

B.l B. 2 B.3 B. 4 B.5

B. Torsion . . . . Geometrical Definition of Torsion Geometrical Properties of Torsion Algebraic Definition of Torsion . Torsion and Polyhedra . . . . . Torsion and Homotopy Equivalences.

93 94

96 97 97

98 99

100 100 100 101 102

104 104 104 106 106 107

Historical Notes

108

Bibliography .

112

Index . . . .

119

Chapter 1. Polyhedra and P.L. Maps

In this chapter we introduce the main objects of study, polyhedra and p.l. maps. The chapter consists mostly of definitions, examples, and exercises. In a final section we introduce the main results of the book: the Poincare conjecture and the h-cobordism theorem. This section may be omitted until after Chapter 5 if the reader wishes; we have included it here to give a taste of deeper results. Basic Notation

A map is a continuous function. cl(X) denotes the closure of X. IR denotes the real numbers and IRn (Euclidean n-space) the space of n-vectors {x=(x1 , x 2 ' " ' ' xn)} ofreal numbers. We will use the product metric on IRn given by d(x, y)=sup IXi- yd. "Linear" always means linear in the affine sense; thus a linear subspace (or just subspace) V c IRn is a translated vector subspace, or equivalently: for each finite set {ail c V and real numbers Ai with LAi=l we have LAiaiEV. A map f: V_IRm is linear if f(L Ai a i) = L Ai f(aJ Joins and Cones Let A, Be IRn be subsets. Define their join AB to be the subset AB = {A a+,u blaEA, pEB} where A, ,uEIR, A, ,u~O and A+,u= 1. Then AB consists of all points on straight-line segments "arcs" with endpoints in each of A and B. If A = Ii> we define AB = B.

B

Fig.l

AB

2

Chapter 1. Polyhedra and P. L. Maps

If A={a} is a one-point set then we often abbreviate {a} to a. We say that aB is a cone with vertex a and base B (or simply that aB is a cone) if each point not equal to a is expressed uniquely as Aa + pb with b e B, A, p ~ 0 and A + p = 1. Equivalently a fj; B and the arcs abl and aba, for each pair of distinct points bl , b 2 e B, meet only at a. Example

Fig. 2

aB1 is a cone while aB2 is not. The example makes it clear that the property of being a cone depends on the presentation of the set aBo

Polyhedra 1.1 A subset PclRn is a polyhedron if each point aeP has a cone neighbourhood N=aLin P, where Lis compact; N is called a star of a in P and L a link and we write N =N,,(P), L=L,,(P). Note that the case L=~ is not excluded so that a point is a polyhedron. Examples of Polyhedra

/

/

/

/

/

/

/

/

Fig. 3. A house with 2 rooms, each having one entrance

3

Polyhedra

Fig. 4. A pyramid with a flag sitting on an infinite plane

Examples of Non-Polyhedra

a

Fig. 5

x = an open disc with a tail

x = an open square with a tail

In ,the first e~ampl a has no cone neighbourhood in C. In the second example a has a cone neighbourhood aLc:.X but L is non-compact. However X -a is a polyhedron! More examples are given in 1.3, below.

1.2 Remark. In 1.1 we could take N to be the e-neighbourhood N.(a,P)={xlxeP, d(a,x)~e} and Lto be N(a,P)={xlxeP, d(a,x)=/l} for some suitably smalle>O. For given any cone neighbourhood N =aL of a in P, use compactness of L to find an 81 > 0 such that dCa, L) ~ 81, and use the fact that aL is a neighbourhood of x to find an 82 > 0 such that Nez (a, P) c:. aL. Then if 8 = min (810 82) it is easy to see that Ne (a, P) = aNs (a, P) is a cone (Fig. 6).

4

Chapter 1. Polyhedra and P. L. Maps

Fig. 6

1.3 Examples and exercises (1) JR.n is a polyhedron. The subset JR.:cJR.n, defined by xn~O, is a polyhedron. Subspaces of JR.n are polyhedra. (2) An open subset of a polyhedron is a polyhedron. (3) The intersection of finitely many polyhedra is a polyhedron. (Use 1.2.) (4) Let Ir,~cJR.nm be polyhedra and identify JR.nxJR.m with JR.n+m by (x, y) 1-+ (xl' ... , x n, Yl' ... , Ym) then Ir x Pz c JR.n+m is a polyhedron. For if a1 L 1 , a 2 L2 are cone e-neighbourhoods then so is a1 Ll x a 2 L 2 • (5) Let P= l! where P, c JR.n are compact polyhedra and the union is locally finite in the sense that each point pE P has a neighbourhood meeting only finitely many of the l!. Then P is a polyhedron. (Use 1.2.) (6) Cubes. Let a = (al' ... , an)E JR.n. Then N.(a, JR.n) = [a 1 - e, a1 + e] x ... x [an-e, an+e] is a polyhedron by (4), called a "cube". Aface of N.(a, JR.n) is obtained by replacing each factor [ai-e, ai+e] either by itself or by {ai-e} or {ai+e}, and then the faces are also polyhedra by (4) and hence N.(a, JR.n) which is the union of the proper faces (i.e. the faces not equal to the cube) is a polyhedron by (5). We write r for the unit n-cube [ -1, 1]n = Nl (0, JR.n) and jn = Nl (0, JR.n) for its boundary. ]1 = [ -1,1] c JR. should not be confused with the unit interval ] = [0, 1] c JR.. (7) A cone aP on a compact polyhedron P is itself a compact polyhedron. For let xEaP, then if x=a we can take Nx(aP)=aP and if X9=a yve can take Nx(aP)=aNy(P) where x=.il.a+J.lY, YEP; since we have aNy(P)=x(Ny(P)uaLy(P») when X9= y, and aNy(P)= y(aLy(P») when X= y. See Fig. 7. (8) Suppose Pc V is a polyhedron in a subspace and f: V -4 JR.m is linear and injective then f(P) is a polyhedron.

U

Piecewise-Linear Maps

5

Fig. 7

By 1.2 and examples (3) and (6) we can assume that all links and stars are polyhedra. This we do from now on. Piecewise-Linear Maps 1.4 A map f: P ---> Q between polyhedra is piecewise-linear (abbreviated p.l.) if each point aEP has a star N =aL such that f(A a + J1 x) = Af(a) + J1f(x) where XEL and A, J1~O, A+ J1= 1. In other words, f is locally conical, in the sense that it maps rays of the local cone structure linearly.

1.5 Examples (1) A linear map is p.l. (2) The restriction of a p.l. map to a subpolyhedron is p.l. A subpolyhedron is a subset which is itself a polyhedron. (3) Define f: P ---> Q to be linear if it is the restriction of a linear map IR n---> IRm. Then, combining (1) and (2),f is p.l. (4) Let P = U~ be a locally finite decomposition of P into compact subpolyhedra. Iff: P--->Q is a map such thafl~ is p.l. for each 0:, then f is p.l.

Remark. Combining examples (3) and (4), we see that a map which is linear in pieces is p.l. In Chapter 2 we prove that all p.l. maps are obtained in this way, and this explains the terminology. 1.6 Exercises (1) The cartesian product of two p.l. maps is p.l. (2) The composition of two p.l. maps is p.l. (3) The cone construction. Let aP, bQ be cones and f: P ---> Q a map.

6

Chapter 1. Polyhedra and P. L. Maps

Define the cone on!, 1': aP-+bQ by f'(la+J.tx)=lb+J.tf(x) where xeP. Prove that the cone on a p.l. map or homeomorphism is itself a p.l. map or homeomorphism, provided that P and Q are compact. (4) A map f: P -+ Q is p.l. if and only if the graph off r(f)= {(x,f(x»)eJRft+m IxeP}

is a polyhedron. Hint: l(x,f(x»)+J.t(Y,f(y»)=(z,f(z») for some z if and only if f(l x + J.t y) = If(x) + J.tf(y)·

(5) Show that the inverse of a p.l. homeomorphism is again p.l. P.l. homeomorphism is the principal equivalence relation of p.l. topology, and properties preserved under p.l. homeomorphism are called p.l invariants. We will often use the symbol ~ for a p.l. homeomorphism. 1.7

Exercises

(1) Give examples to show (a) The union of two polyhedra is not necessarily a polyhedron. (b) The infinite union of compact polyhedra is not necessarily a polyhedron. (c) The image of a non-compact polyhedron under an injective p.l. map need not be a polyhedron. What about compact polyhedra. and general p.l. maps? (See 2.5 for answers.) (2) Show by radial projection that the (topological) homeomorphism class of La(P) is a p.l. invariant of the pair (a, Pl.

The Standard Mistake The last exercise prompts the observation that projection maps are not necessarily p.l. For example the graph of a projection of one arc into another is part of a hyperbola.

b' --------------b

o 0' --

0'

o

b'

Fig.S

Manifolds

7

A p.l. version of exercise (2) will be given in Chapter 2, using "pseudoradial projection".

P.L. Embeddings Exercise 1.7 (c) shows that we have to be careful about defining p.l. embeddings. We say that a p.l. mapf: P - Q is a pJ. embedding provided f(P) is a subpolyhedron of Q and f: P - f(P) a p.l. homeomorphism. Convention. From now on we will usually omit the prefix p.l.

Thus map, embedding, homeomorphism will mean p.l. map etc. When we have need to use non p.l. maps we will use the phrase "topological map" fit order to avoid confusion.

Manifolds

1.8 A polyhedron M is an unbounded p.l. manifold of dimension n (or simply an n-manifold) if each point xeM has a neighbourhood in M, which is (p.l.) homeomorphic to an open set in R"; such a neighbourhood is called a coordinate neighbourhood. We often indicate the dimension of an n-manifold M by writing M". M is an n-manifold with boundary if each point has a neighbourhood homeomorphic to an open subset of either R" or R~. Define the boundary of M, aM, an unbounded (n-1)-manifold, to consist of points The boundary is well-defined by corresponding to R" -1 X 0 c: R~. 1.7 (2) and elementary algebraic topology. This also follows by an easy induction using p.l. invariance of links (2.21 (2»). Terminology. A manifold M is closed provided oM =~ and M is compact. If M is any manifold, define the interior of M, intM, to be M-oM.

1.9 Examples and exercises (1) R", R~ and subspaces of R" are manifolds. (2) An open subset of a manifold is a manifold. (3) The product of an n-manifold with a q-manifold is an (n+q)manifold. Hint: Derme a homeomorphism of R!+={xe2,l~OX} onto R! by using a linear homeomorphism of R!++={xeR 2, Xl ~ x 2 ~ O} onto R! +. Use this on suitable coordinates to derme a x R~ onto R~+q. homeomorphism of R~

8

Chapter 1. Polyhedra and P.L. Maps

••

-

""::".

IR+~

Fig. 9

(4) It follows from (3) that 1" is an n-manifold with boundary. (5) IR. n ~ a1"+ 1 - one point. (This is difficult, see 3.20 for a proof using machinery.) Balls and Spheres

A manifold homeomorphic with 1" is called an n-ball or n-disc often written B n or Dn. A manifold homeomorphic with a1"+l is called an n-sphere, usually written sn. 1.10 Lemma. Let Bn, Dn be n-balls and h: aw --4 aDn a homeomorphism. Then h extends to a homeomorphism hi of B n with Dn. Proof We can assume Bn=Dn=1" and then define hdAx)=Ah(x) for xEi n and O~A 1. This is the cone construction applied to 1"=ojn.

--h

Fig. to

The P,oincare Conjecture and the h-Cobordism Theorem

We now state the main theorems for which we are heading. Poincare conjecture. Let M n be a closed manifold having the homotopy type of an n-sphere, then M is an n-sphere.

The Poincare Conjecture and the h-Cobordism Theorem

9

Theorem A. The conjecture is true for n ~ 6.

In fact the conjecture is true for n = 5, but the proof at the moment is beyond the scope of an elementary treatment. For n = 3, 4 the conjecture is still, at the time of writting, unsolved. We will deduce Theorem A from the h-cobordism theorem (Theorem B below). A cobordism (WW, M o , M1) consists of a compact manifold W with oW the disjoint union of manifolds Mo and MI' When Mo and MI are understood, we refer to W itself as a cobordism. W is an h-cobordism if both inclusions Mo c: Wand MI c: Ware homotopy equivalences. Theorem B. Suppose WW is a simply connected h-cobordism and w ~ 6. Then W~MoxI and hence Mo~I' Remark. If M o , MI and Ware all simply-connected, then by Whitehead's theorem (see Appendix A) it is enough to assume that all the relative homology groups H* (w, Mo) and H* (w, M 1) vanish. But by Lefschetz duality (see appendix and proof given in Chapter 6) it is enough to assume this for one end only. Consequently we can state Theorem B in the following form, which is the form in which it will be proved.

Theorem B/. Suppose (WW, M o , M 1) is a cobordism and that (1) nl(Mo)=nl(MI)=nl(W)=O (2) H*(W, Mo)=O (3)

w~6.

Then W~Mo

x I.

We shall al~o prove a relative version of the theorem (for cobordisms between manifolds with boundary) and a version for non-simply

Fig.!!

10

Chapter 1. Polyhedra and P. L. Maps

connected manifolds (the s-cobordism theorem). We conclude this chapter by showing that Theorem A follows from Theorem B': In M choose two disjoint standard n-cubes inside coordinate neighbourhoods. Call them Dl and D 2 . Denote ~=cl(M-D) and W=cl(~-D2)' Then ~ and Ware manifolds since cl(JRn-In) is a manifold by an exercise on the lines of 1.9(3). We claim that W is an h-cobordism between oD I and oD2. First of all 1tl (oD 1) = 1t1 (oD 2) =0 and 1t1 (W)=1t 1 (M)=0 since W has the homotopy type of M - {two points}. ' Now (excision)

H.(W, OD2)~H.(,

~i.()

(since D2 is contractible).

But " H*(~)n-.,

oD.) Dt ) ~Hn-·(M,

(Lefschetz duality) (excision)

~in-*(M) ~

{ 071.

* =0 . M IS . a h omotopy sphere. h . }slDce ot erwlse

It follows that i.(~)=o and hence that W is an h-cobordism. By Theorem B' there is a homeomorphism h: W -+ jn X II and we extend h to a homeomorphism of M with jn+! by two applications of 1.10.

Chapter 2. Complexes

In this chapter we introduce the principal tools of p.l. topology: simplicial complexes and simplicial maps. The connections between these and polyhedra and p.l. maps is the major concern of the chapter. The rest of the chapter deals with other useful tools: pseudo-radial projection, joins and collars. The results on convex cells which we need are given in an appendix to the chapter. Simplexes 2.1

Proposition. The join operation is associative and commutative and

Proof Define Ao At ... An inductively to be (Ao ... A n- t ) An and prove the identity inductively. Associativity and commutativity then follow. The induction step follows from the equation

Now define a finite set {vo,vt, ... ,vn}c]Rm to be independent if it is not contained in any subspace of dimension < n, or equivalently if the vectors {Vi - vol are linearly independent. Then define an n-simplex A c JRm to be the repeated join Vo VI ..• L'n of n + 1 independent points. We call the points Vi the vertices of A and say that they span A. A simplex spanned by a subset of the vertices is called a face of A. If B is a face of A we write B 2. Ann. of Math. 83, 402-436 (1966). Rourke, C.P.. Sanderson, B.I.: Block bundles, III. Ann. of Math. 87, 431-483 (1968).

P. Immersion theory P.1 Haefliger, A., Poenaru, V.: La classification des immersions combinatoire. Publ. I.H.E.S. (Paris) 23, 75-91 (1964). Differential case: P.2 Smale, S.: The classification of immersions of spheres in euclidean space. Ann. of Math. 69, 321-344 (1959). P.3 Hirsch, M. W.: Immersions of manifolds. Trans. Amer. Math. Soc. 93, 242-276 (1959). Q. SmQothing theory Q.1 Whitehead,I.H.C.: On C1-complexes. Ann. of Math. 41, 809-814 (1940). Q.2 Cairns, S.S.: A simple triangulation method for smooth manifolds. Bull. Amer. Math. Soc. 67, 389-390 (1961). Q.3 Munkres,I.: Obstructions to smoothing piecewise differentiable homeomorphisms. Ann. of Math. 72, 521-544 (1960). Q.4 Mazur, B., Hirsch, M. W.: Obstruction theories for smoothing manifolds and maps. Bull. Amer. Math. Soc. 69, 352-356 (1963). Q.5 Kervaire, M.A.: A manifold which does not admit any differentiable structure. Comm. Math. Helv. 34, 257-270 (1960). Kervaire and Milnor [N.3l. Q.6 Lashof, R., Rothenberg, M.: Microbundles and smoothing. Topology 3, 357-388 (1965).

118

Bibliography

Haefliger, A.: Lissages des immersions, I. Topology 6, 221-240 (1967). Haefliger [0.14). Rourke and Sanderson [1..1J, Section 6. Q.8 Cerr,J.: Lanullitedelto(DiffS3).Seminar Cartan 1962/3, expo 8,9,10,21 and 22. Q.9 Rourke, C. P.: Structure theorems (to appear). Q.7

R. Triangulation of topological manifolds and the hauptvermutung

R.l R.2

R.3 R.4 R.5

R.6

Dimension 3: Moise, E.E.: Affine structures on 3-manifolds. Ann. of Math. 54-59 (1951-1954). Bing, R. H.: An alternative proof that 3-manifolds can be triangulated. Ann. of Math. 69, 37-65 (1959). . Dimension ~ 5: Milnor,J.W.: Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. 74, 575-590 (1961). Kirby, R., Siebenmann, L.: On the triangulation of manifolds and the hauptvermutung. Bull. Amer. Math. Soc. 75, 742-749 (1969). Kirby, R.C., Siebenmann, L.C.: Foundational essays in topological manifolds, sfuoothings and triangulations. Ann. of Math. Study number 88, Princeton U.P. (1968). Rourke [Q.9). Siebenmann, L.C.: The disruption of low-dimensional handlebody theory by R.ohlin's theorem. Topology of manifolds, Edit. by Cantrell and Edwards, p.57-76. Chicago: Markham 1970. Rourke and Sanderson [L.ll].

Homotopy hauptvermutung: Casson [N.ll), R.7 Sullivan,D.P.: On the hauptvermutung for manifolds. Bull. Amer. Math. Soc. 73, 598-600 (1967). R.8 Armstrong, M.A., Cooke, G.E., Rourke, C.P.: Princeton notes on the hauptvermutung (1968), available from Warwick Univ. (Coventry). S. Bordism and Cobordism Differential case: S.l Thom, R.: Quelques proprietes globales des varietes differentiables. Comm. Math. Helv.2I, 17-86 (1954). S.2 Wall,C.T.C.: Determination of the cobordism ring. Ann. of Math. 72, 292-311 (1960). S.3 Stong, R.E.: Notes of cobordism theory. Princeton Math. Notes Series number 7, Princeton U.P. (1968). S.4 Ati)'ah, M.F.: Bordism and cobordism. Proc. Cambridge Philos. Soc. 57, 200-208 (1961). P.1. case: S.5 Williamson,R.E.,Jr.: Cobordism of combinatorial manifolds. Ann. of Math. 83, 1-33 (1966). S.6 Browder, W.. Luilevicious, A., Peterson, F. P.: Cobordism theories. Ann. of Math. 84,91-101 (1966). S.7 Brumfiel, G., Milgram, J., Madsen, I.: P.l. characteristic classes and cobordism. Bull. Amer. Math. Soc. 77, 1025-1030 (1971). Rourke and Sanderson [J.4).

Index

Abstract isomorphism 20 - polyhedron 26 - simplicial complex 26 Adding handles 80 - lemma 80 Alexander trick 37 - duality 100 Ambient isotopy 37 Annulus theorem 36 Arc 1 Attaching sphere and tube of handle 74 - map of handle 74 - cells 101 Ball 8 - complex 27 -, joins of 23 -, pairs of 50 -, joins of pairs of 52 -, unknotting pairs of in codimens'~3 91-92 - - - - in codimension 2 92 Barycentre of simplex 11 Based handle 89 Basepoint 100 Belt sphere and tube of handle 74 Bordism, bibliography 118 Boundary of cube 4 - of manifold 7' - of simplex 12 - of cell 13 - of cobordism 87 Cancelling handles 78 Cancellation lemma 78 Cell 13 - complex 14 - isa ball 21 -,convex 13,27-30 -, pairs of 51 - in CW complex 101

Cellular map 16 - isomorphism 16 - collapse 104 - expansion 104 - moves 54-55, 65 Characteristic map for handle 74 - - for cell 101 Closed map 60 - manifold 7 Cobordism 9 -, handles on 75 - with boundary 87 -, invertible 93 -, bibliography 118 Cocore of handle 74 Codimension 50 Coefficients 100 Cohen's simplicial neighbourhood theorem 34-35 Cohomology 100 Collapse and collapsing 39 et seq, - for pairs 54 -, cellular 104 -,internal 105 -, polyhedral 105 -, collapsing and regular neighbourhoods 40 -, notes 109-110 Collar and collaring 24 -,local 24 - theorem 24 - for pairs 52 collars as regular neighbourhoods 36 -, regular neighbourhood collaring theorem 36 Combinatorial annulus theorem 36 - manifold, notes 108 Complement, simplicial 32 Complementary handles 78 Complex, cell 14 -, simplicial 16

Index

120 Complex, ball 27 -,dual 27 -,CW 101 Computing homology 102 Concordance of embeddings 96 Cone 2 - construction 5-6 - on p.l. map 6 - on complex 15 -,dual 27 - pair 48 Connected sum 46 Connectivity, simple and higher 101 Constructing h-cobordisms 90 Contractible 98 Convex set 13 - cells 13, 27-30 Coordinate neighbourhood 7 Core of handle 74 Critical dimension for linking 69 Cube 4 Cycle 98 CW complex 101 - associated to a decomposition 83 -, subdivision of 105 Decomposition (handle) 81 -, symmetrical 82 -,nice 82 -, associated CW complex 83 -, simplifying 84 Deformation retract 98 Derived·subdivision 20 -, near subcomplex 32 - neighbourhood 32 - neigbbourhood for pairs 52 Diagram of maps 18 Dimension 12 Disc 8 - theorem 44 - theotem for pairs 56 Dual cone 27 - complex 27, 84 Duality theorems 84,100 Elimination of handles 85-86 Embedding in double dimension (Penrose-Whitehead-Zeeman theorem) 63 - in codimens~3 (Irwin's theorem) 96 -, notes 111 -, bibliography 113, 116 Engulfmg 94

-:-' bibliography 114 Epsilon (s-) neighbourhood 32 - homotopy 60 - isotopy 60 Euclidean space 1 Exactness 97 Excision 98 Expansion (cellular) 104 Extending collars 57 External join 22-23 Face of cube 4 - of simplex 11 - of cell 14 Five dimensional theorems 93-94 Foundations, bibliography 112 Full subcomplex 31 Fundamental group 101 Geometric interpretation of homology 98 Geometric topology, notes 108 General position 60-63, 64 - - theorem for embeddings 61 - - theorem for maps 61 - -, bibliography 113 Gluing 26 h-cobordism 9 - theorem 9 -, proof Qf theorem 87 -, relative theorem. 87 -, classification 90 -, construction 90 -, weak five dimensional theorem 93 -, notes 109 Handle 74 et seq. -, terminology 74 - on cobordism 75 -, reordering 76 - of adjacent index 76 -, incidence number of 77 -, complementary 78 -, cancelling 78 -, introduction 79 -,adding 80 - deqomposition 81 -, nice decomposition 82 -, eliminating 85-86 -, based 89 -, bibliography 114 Hauptvermutung, bibliography 118 Homeomorphism, p.l. 6 -, periodic 26

Index

121

Homogeneity of manifolds 44 Homology 97 - triviality of links 69-70 - linking number 72 -, geometric interpretation of 98 - between cycles 99 - of sphere 99 -, computation of 102 -, bibliography 115 Homotopy 97 -,Il- 60 - equivalence 98 - groups 100 -, simple 39,107 House with two rooms 2, 40 -, notes 108 Immersion theory, bibliography 117 Independent set 11 - subsets 22 Index of handle 74 Induced orientation for boundary of manifold 45 Incidence number, of handle 77 - - of cell 102 - - in Z1t 89,103 Interior of manifold 7 - of cell 13 Internal collapse 105 Intersection number 68, 100 - - in Z1t 72 Introducing handles, introduction lemma 79 Invertible cobordism 93 Irwin's embedding theorem 96 Isomorphism, cellular 16 -, abstract 20 Isotopy 37 -, ambient 37 -, support of 37 - uniqueness of regular neighbourhoods 38 - extension 56-59 -, locally trivial 58 - extension theorem 58 -,Il- 60 -, notes 110 -, bibliography 113 Join 1 -, external 22-23 -, simplicial 23 - of balls and spheres

23

- of maps 23 - of pairs of balls and spheres -, bibliography 112

51

Lefschetz duality 84,100 Level preserving 37 - - lemma 58 Levine's unknotting theorem 92 Linear map 1 - subspacel - cell 13 - triangulation 18 Link 2 - of vertex in simplicial complex - of simplex 23 - pair 48 - of spheres 69 Linking number 72 Local collaring 24 - collaring for pairs 52 - extension of collar 57 - triviality of isotopy 58 - flatness 47, 50

20

Manifold 7 -, homogeneity 44 - pair 50,51 - embedding theorems 63, 96 Map, linear 1 -,p.l. 5 -, cellular 16 -, simplicial 16 Mapping cylinder 107 Morse function, notes 109 Naturality 97 Neighbourhood, simplicial 32 -, derived 32 -,Il- 32 -, regular 33 et seq. Newman's theorem (corollary 3.13) 35 - -, notes 109 Nice handle decomposition 82 Non-degenerate map 61 Normal bundles, bibliography 115 Orientation of manifold 43 et seq. -, definition independent of algebraic topology 46 - induced on boundary 45 -, standard orientation for spheres and balls 45 -, oriented cycle 98

122 P.l.map 5 homeomorphism 6 - invariant 6 - embedding 7 - manifold 7 -, notes 108 Pairs 50 et seq. Periodic homeomorphism 26 Piecewise-linear, see p.l. Piping 67 Poincare conjecture 8 - theorem (dimension!1;6) 8 -, weak 5-dimensional theorem 94 - duality 84, 100 -, notes 108-109 -, bibliography 114 Polyhedral collapse 105 Polyhedron 2 -, examples 2 -, non-examples 2 -, abstract 26 -, pairs of 50 -, notes 108 Projection, radial 6 Proper manifold pair 50 Pseudo-radial projection 20-21 Radial projection 6 Realising abstract simplicial complexes 26 Reduction of collar 57 Regular neighbourhood 33 et seq. - uniqueness theorem 33 - in manifolds 34 - collaring theorem 36 -, isotopy uniqueness (regular neighbourhood theorem) 38 -, collapsing criterion 41 - for pairs 52 - theorem for pairs 53 - -, notes 109 - -, bibliography 112 Relative h-cobordism theorem 87 Relative regular neighbourhoods 56 - - -, notes 109 - - -, bibliography 112 Reordering handles and the reordering lemma 76 s-cobordism theorem 88 et seq. - -, notes 109 - -, bibliography 114 Schonflies conjecture 47,50

Index -, weak theorem 47 -, notes 110 -, bibliography 113 Shelling 40 - for pairs 54 Sign of intersection 68 Simple connectivity 101 Simple homotopy, equivalence, type 39, 107 - _, bibliography 113 Simplex 11 -, vertex of 11 -, face of 11 -, boundary of 12 Simplicial complex 16 - map 16 - diagram 18 - join 23 -, abstract complex 26 - neighbourhood 32 - complement 32 - collapsing, notes 109 Simplicial neighbourhood theorem 34-35 - - - for pairs 53 Singular set 60 - cycle 99 Singularity of cycle 98 Skeleton of complex 15 Smale's h-cobordism theorem, see h-cobordism theorem Smoothing theory, bibliography 117 Spanning of simplex 11 - subspace 12 - of cell 13 Sphere 8 -, joins of 23 -, standard orientation of 45 -, pairs of 50 -, criterion for unknotting 55 -, unknotting 64,91-92 -, link of 69 -, homology of 99 Standard mistake 6, 108 - orientations 45 - ball and sphere pairs SO - link 69 Star 2 - in complex 15 - pair 48 Starring, stellar subdivision, stellar moves 15 -, notes 109 Subcomplex 15

123

Index - full 31 Subdivision IS -, stellar IS - of triangulation 18 -, derived 20 - of CW complex 105 - lemmas 16, 17, 19, 31 - theorems 17,18 (for trees) - counterexamples 19 Support of isotopy 37 Surgery, bibliography 116 Symmetrical handle decomposition

-

82

Torsion (Whitehead torsion) 40,104-107 - and s-cobordism theorem 88 et seq. -, constructing h-cobordisms with given 90 -, geometric definition and properties 104 et seq. -, algebraic defmition 106 -, connection with polyhedra 106 -, connection with homotopy equivalences 40,107 -, bibliography 115 Trail 40 Transversality 61 -, bibliography 115 Tree 18 Triangulation 17 -, subdivision of 18

of topological manifold, bibliography 118

Underlying polyhedron to cell complex 14 Unit interval 4 Universal cover 102 - coefficient theorem 100 Unknotted ball and sphere pairs SO Unknotting theorems for balls and spheres in codimension ~3 (Zeeman's theorem) 50,52, 55, 64, 91-92 - - - - codimension 2 (Levine's theorem) 92 - -, notes 111 - -, bibliography 113, 116 Unlinking spheres 69, 70 Vertex ofsimplex - of cell 14

11

Weak five dimensional theorems 93-94 Whitehead group ofspace 104 - - of group 106 Whitehead torsion, see torsion Whitehead's theorem 102 Whitney lemma 68, 69, 78 - - non-simply-connected case 72 - - notes 110 Zeeman's theorem, see unknotting

R Bott, L. W. Tu

W.S.Massey

DifJerentiai Forms in Algebraic Topology

Algebraic Topology: An Introduction

1981. Approx. 83 figures. Approx. 250 pages (Graduate Texts in Mathematics, Volume 82) ISBN 3-540-90613-4 Thisauthoritativetext-developedfromafirstyear graduate course in algebraic topology - is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. Its unique approach, using the de Rham theory of differential forms as a prototype of a cohomology theory, makes the entire field easier to assimilate.

5th corrected printing. 1981. 61 figures. XXI, 261 pages (Graduate Texts in Mathematics, Volume 56) ISBN 3-540-90271-6 Professor Massey's well-known and popular text is intended as an introduction to algebraic topology for advanced undergraduate and beginning graduate students. The standard topics - two-dimensional manifolds, the fundamental group, covering spaces, and group theory - are covered. They receive straightforward treatmentstrippedofunnecessary definitions and terminology. The interplay stressed between algebra and topology, causing each to reinforce interpretations ofthe other, helps to emphasize the essential unity of all mathematics.

R H. Crowell, R H. Fox

Introduction to Knot Theory 4th corrected printing. 1977.65 figures. X, 182 pages (Graduate Texts in Mathematics, Volume 57) ISBN 3-540-90272-4 Contents: Knots and Knot Types. - The Fundamental Group. - The Free Groups. Presentation of Groups. - Calculation of Fundamental Groups. - Presentation of a Knot Group. - The Free Calculus and the Elementary Ideals. - The Knot Polynomials. Characteristic Properties ofthe Knot Polynomials. IL.Kelley

General Topology Unchanged reprint. 1975. XIV, 298 pages (Graduate Texts in Mathematics, Volume 27) ISBN 3-540-90125-6 . Distribution rights for Japan: Toppan Co. Ltd., Tokyo

W.S.Massey

Singular Homology Theory 1980. 13 figures. XII, 265 pages (Graduate Texts in Mathematics, Volume 70) ISBN 3-540-90456-5 This book is a textbook on homology and c0homology theory for students at the beginning graduate level The singular homology theory is developed systematically starting from the scratch. The only formal prerequisites are a basic knowledge of abelian groups and point set topology. The book is a continuation ofthe author's earlier book Algebraic Topology: An Introductionwhim contains important supplementarymaterial sum as the theory ofthe fundamental group and a discussion of2-dimensional manifolds. The presentation is straightforward, avoiding all unnecessary defmitions, terminology, and tecftnica1 machinery. The geometric motivation behind the various algebraic concepts is emphasized where possible.

Springer-Verlag Berlin Heidelberg New York

E.E.Moise

1 Stillwell

Geometric Topology in Dimensions 2 and 3

Oassical Topology and Combinatorial Group Theory

1977. 92 figures. X, 262 pages (Graduate Texts in Mathematics, Volume 47) ISBN 3-540-90220-1

Illustrated with 305 figures by the Author 1980.305 figures. XII, 301 pages (Graduate Texts in Mathematics, Volume 72) ISBN 3-540-90516-2 This is a unique book, full of intriguing insights into a complicated and interesting subject matter. Conceived in the tradition of Seifert & Threlfall, it confines itself to dimensions ~ 3 with many results on groups, 2-manifolds and 3-manifolds.There are many historical remarks; the most outstanding feature of the book, however, is its more than 300 illustrations. The author has taken great care to rresent his text in a readable way and includes many interesting exercises and challenging open problems.

L.ASteen, lASeebach, Jr.

Counterexamples in Topology 2nd printing of the 2nd edition. 1981. 31 figures, 6 tables. XI, 244 pages ISBN 3-540-90312-7 Counterexamples in Topology is a compen150 significant examples of dium of n~ar1y topological spaces, each discussed in detail. Examples range from the trivial to the unbelievable, from the well known to the obscure. The examples are preceded by a succinct exposition of general topology which establishes terminology and outlines the basic theory. Reference charts at the end summarize the properties of the various examples. Additions to the second edition include up-todate references and bibliographical notes concerning recent refinements in examples. An entire new section has been added containing a revised and updated version of the first author's award-winning paper "Conjectures and Counterexamples in Metrization Theory", which first appeared in 1972 in American Mathematical Monthly. Some of the special features are the book's emphasis on the geometrical and intuitive side of the discussed examples, an unusually complete treatment of order spaces, paracompactness and metrization theory, and an extensive collection of exercises correlated with the various examples.

G. W Whitehead

Elements of Homotopy Theory 1978.46 figures, 1 table. XXIII, 744 pages (Graduate Texts in Mathematics, Volume 61) ISBN 3-540-90336-4 This book examines the elementary portions of homotopy theory. It is assumed that the student is familiar with certain basic concepts, like the fundamental group and singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential to an understanding ot this concrete and careful text. A special feature of this introduction is the use of elementary methods whenever possible. This approach entails a leisurely exposition in which brevity and, perhaps, elegance are sacrificed in favor of concreteness and ease of application. The book also makes homotopy theory accessible to readers involved in a wide range of other subjects subjects in which its impact is beginning to be felt.

Springer-Verlag Berlin Heidelberg New York