Combinatorial Topology Volume 3 [03] 0486401790, 9780486401799

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P.S. Aleksandrov

COMBINATORIAL TOPOLOGY Volume 3

COMBINATORIAL TOPOLOGY VOLUME 3

OTHER GRAY LOCK PUBLICATIONS

KH INCHIN: Three Pearls of X umber Theory

Mathematical Foundations of Quantum Statistics PONTRYAC.IN: Foundations of Combinatorial 'Topology

Differentiable Manifolds and Their Applications to Homotopg Theory NOYOZII ILOY: Foundations of the Nonlinear Theory of

Elasticity KOLMOCOROY unci HOMIN': Elements of the Theory of Functions and

Functional Analysis. Vol. 1: Metric and Xonncd Spaces PETROYSKII: Lectures on the Theory of Integral Equations ALEKSANDROV: Combinatorial Topology

Vol. 1: Introduction. Complexes. Coccrings. Dimension Vol. 2: The Betti Groups

COMBINATORIAL TOPOLOGY VOLUME 3

BY

P. S. ALEKSANDROV

G R A P L 0 C K

R R E S S

ALBANY, N. V. 1900

TRANSLATED FROM THE FIRST (1947) RUSSIAN EDITION BY HORACE KOMiM

Copjudght, 1960, by GRAYLOCK PRESS Albany, N. Y.

All rights reserved. This book, or parts thereof, may not be reproduced in any form, or translated, without permis­ sion in writing from the publishers.

Library of Congress Catalog Card Xumber 56-13930

Manufactured in the United Stales of America

CONTENTS PART FOUR HOMOLOGICAL MANIFOLDS. TIIK DUALITY THEOREMS. COHOMOLOGY GROUPS OF COMPACTA Chapter

XIII. H omological M anifolds (/(-M anifolds ) 1. Definition and Simplest Properties........................ 1.1. Definition of/(.-manifolds.................................................. 1.2. Elementary properties of //-manifolds............................ 1.3. The case n < 3 .................................................................. 1.4. Barycentric stars and/i-manifolds.................................. 2. The Barycentric Complex of a Combinatorial A-Manifold . 2.1. Notation; preliminary basic facts.................................... 2.2. The complex AC.................................................................. 3. The Intersection Number, the Isomorphism Dq and the Poincare Duality........................................................................ 3.1. The intersection number { i f X *tq) .............................. 3.2. The intersection number and the incidencenumbers. . 3.3. The isomorphism Dq and the Poincare duality.......... 3.4. The intersection number (.rp X *xq) .............................. 3.5. Veblen’s theorem................................................................ 4. The Combinatorial Form of the Alexander Duality.......... 4.1. Formulation of the theorem............................................. 4.2. Generalization of Theorem 4.10...................................... 4.3. The cases p = 0 and p = n — 1; prefatory remarks to the proof of 'Theorem 4 .2 ................................................. 4.4. The isomorphism ADq...................................................... 4.5. Definition and simplest properties of the linking num­ ber v(up, u"-1) .................................................................... 4.G. Theorem on linked systems of cycles.............................. X IV . C ohomology G roups P oxtryagix D uality

of

C ompacta

and the

11 11 14 17 21 23 24 24 20 29 30 34 37

A lexaxder -

1. Cohomology Groups of Bicompact a ........................................ 1.1 Spectra and their cohomology groups.............................. 1.2. The homomorphism Sa ..................................................... 1.3. The cohomology groups of bicompacta.......................... 1.4. The group V00...................................................................... 2. Cofinal Suborders of Spectra. The Case of Compacta......... 2.1. Cofinal suborders of spectra............................................. 2.2. The case of T a eompactum.............................................. 2.3. The case of a polyhedron.............................................. V

4 4 4 (i 8 t) !) 10

41 41 44 44 45 40 40 47 48

CONTENTS

VI Chapter

3. The Homology Groups of Open Subsets of S". Formulation of the Alexander-Pontryagin Duality and Its Fundamen­ tal Consequences. Finite Compacta........................................ 3.1. The group A’(P), where r is an open subsetof S'1. . . . 3.2. The barycenfric subdivision of a chain of T.................. 3.3. The topological invariance of the groups A '( r ) .......... 3.4. The Alexander-Pontryagin duality and its fundamen­ tal consequences................................................................. 3.5. Special case: is finite...................................................... 4. The Complexes K a , K 0a , K f , Koa*, Qai ; the Sets Ga and r a........................................................................................ 4.1. Definitions and notation................................................... 4.2. The continuous mapping C of Ga onto j] Qai ||.............. 4.3. Some properties of the group Z' ( F ) ................................ 5. A Combinatorial Lemma.......................................................... 5.1. Introductory remarks and formulation of the lemma. . 5.2. The fundamental identity................................................. 5.3. The completion of the proof of Lemma 5.1................... G. The Spectrum 3C0 of the Set ................................................. 6.1. The coverings fG and o a................................................... G.2. The spectrum 3C0 of the compaction ........................... 6.3. The projections S j ............................................................ 7. Further Auxiliary Propositions................................................ 7.1. Commutativity of the operators S f and V /f................ 7.2. Corollary to (5.1) and (7 .1 )............................................ 8. Proof of the Alexander-Pontiyagin Duality.......................... 8.1. The homomorphism ADq of V;,(4>) into A9- 1(r ) in­ duced by the homomorphisms AD aq............................... 8.2. The mapping A/V is an isomorphism.............................. 8.3. The isomorphism ADq is onto.......................................... XV. L in k in g . T he L ittle A lexander D uality 0. Introductory Remarks............................................................... 1. The Intersection Number and the Linking Number........... 1.1. The intersection number of two chains xp and if of R". 1.2. Computation formulas for intersection numbers.......... 1.3. Corollaries of Theorem 1.2. Generalization of the defi­ nition of intersection number.......................................... 1.4. The linking number........................................................... 1.5. Computation formulas for the linking numbers............ l.G. The case zp f Z f ( K 0), V "1 £ Z f ~ \ K * ) .................... 2. Linking of Proper Cycles..........................................................

50 50 50 51 52 54 50 50 50 59 GO GO G1 G4 GG GG G7 GS 08 GS GO 70 70 71 71 73 73 73 74 77 80 82 83 S5

vii

CONTEXTS

Chapter

2.1. Definition of linking numbers for proper cycles. . 2.2. Deformation theorem for singular cycles....................... 3. The Little Alexander Duality................................................. 3.1. Introductory remarks and formulation of the theorem. 3.2. Proof of Theorem (3 .1 ).................................................... 3.3. The little Pontryngin duality........................................... 3.4. Closed (n — 1) -dimensional pseudomanifolds in S'1 (n > 2). The Jordan-Brouwer theorem........................ 3.5. Remarks on linking of pseudomanifolds. .

85 87 88 88 81) 1)1 1)3 1)5

PART FIVE INTRODUCTION TO THE THEORY OF CONTINUOUS MAPPINGS OF POLYHEDBA X Y I . T he B rouwer T heory On and o

of

C ontinuous M appings

in

R"

1. Index of a Point Relative to an (n — 1)-Cycle in R n........ 100 1.1. The combinatorial case..................................................... 100 1.2. The index of a point o £ R" relative to a proper cycle 5

2. 3.

4.

o.

.......................................................................................................

101

1.3. The Poincarc-Bohl theorem and Rouche’s theorem. . . 103 1.4. Homologies in R n \ o ...................................................... 103 1.5. The interpretation of the index of a point relative to a cycle as the degree of the central projection of the cycle onto a sphere............................................................ 104 The Existence of Roots...... ...................................................... 105 2.1. Existence theorem for roots.............................................. 105 The Local Degree of a Mapping of an n-ChainInto R n. . 100 3.1. Definition and fundamental properties of the local degree................................................................................... 107 3.2. Simplicial approximations................................................ 107 3.3. Mappings into A” .............................................................. 109 Topological Mappings............................................................... 109 4.1. Theorem 4.1........................................................................ 109 4.2. Invariance theorems.......................................................... 110 Vector Fields and Continuous Mappings............................... I ll 5.1. The connection between vector fields and mappings.. . I ll 5.2. The index of an isolated zero of a vector field.............. 113 5.3. Vector fields on a solid sphere.......................................... 114 5.4. Brouwer’s fixed point theorem for an n-cell.................. 115 5.5. Vector fields on spheres of even dimension.................... 110

viii

CONTENTS

Chapter

5.0. Reflections of spheres; another fixed point theorem . . . 5.7. Exercises.............................................................................. 0. Classification of Continuous .Mappings of an -ra-Sphere Into an n-Spherc................................................................................. 0.1. TIopf’s theorem; introductoryremarks............................. 0.2. Mappings of degree zero................................................... 0.3. Reduction of Ilopf’s theorem to an extension theorem. 0.4. Proof of the extension theorem........................................ XVII. F ixed 1. The 1.1. 1.2. 1.3.

P oints

of

C ontinuous M appings

of

117 118 119 119 122 124 125

P olyhedra

Fixed Point Theorem....................................................... Fixed simplexes.................................................................. Ilopf’s formula..................................................................... The Lefschetz number of a continuous mapping of a polyhedron into itself........................................................ 1.4. Examples............................................................................. 1.5. Some remarks on the Lefschetz number........................ 2. The Index of a Fixed Point...................................................... 2.1. Definition of the index...................................................... 2.2. Some properties of the index............................................ 2.3. Normal, fixed points. Topological invariance of the in­ dex of a fixed point............................................................ 2.4. Fixed points of affine mappings....................................... 3. The Algebraic Number of Fixed Points of a Continuous Mapping of a Polyhedron Into Itself...................................... 3.1. Definition of regular fixed points. Formulation of the fundamental theorem........................................................ 3.2. Generalization of Theorem 3.1......................................... 3.3. Regular fixed points of simplicialmappings.................... 3.4. Reduction of Theorem 3.1 to the approximation theorem................................................................................ 3.5. Proof of the approximation theorem............................... 3.0. Remarks on the algebraic numberof fixed points. . . . R e f e r e n c e s ................................................................................................. I n d e x ............................................................................................................

12S 128 129 131 132 133 134 134 135 L3G 138 13S 13S 139 139 140 141 144 146 147

Part Four

HOMOLOGICAL MANIFOLDS. TIIE DUALITY THEOREMS. THE v-GROUPS (COHOMOLOGY GROUPS) OF COMPACTA Polyliedra occupy an intermediate position in topology. For some purposes, the class of figures encompassed by potyhedra is loo wide; for others, too narrow. It is too wide from the point of view of the wealth of concrete geometric facts and problems which are revealed when one passes from general polyliedra to the special case of manifolds, which have proved to be one of the fundamental and central objects of topological research. On the other hand, the class of polyliedra and their topological images proves to be too narrow and adventitious if one is interested in the more general concepts of modern topology. One then seeks a natural class of figures broad enough to allow these concepts, without ceasing to he geo­ metrical (that is, without losing their original connection with our spatial intuition), to reveal the mathematical essence, in all its fullness and di­ versity, of this set of geometric objects. In short, more general concepts require a wider and more abstract field of application, and it is then neces­ sary to include, at any rate, all the compacts and, apparently, even all the bicompact a (bicompact Hausdorff spaces). We wish to emphasize1 once more, however, that our object is not to generalize for the sake of generali­ zation, but to understand the deeper laws governing a broad, difficult and diverse field of mathematical phenomena. It is therefore not surprising that we discern two clearly distinguished directions in topology: the topology of manifolds and the topology of bi­ compact and locally Incompact Hausdorff spaces. Each of these main fields has many branches; in this book wc can con­ sider only the simplest and most fundamental results of a few of these subfields. We discuss those parts of the theory of manifolds and bicompact spaces which can be treated by homology methods. The rigorous treatment of homology theory was begun in Chapter VII, and the methods introduced there and in subsequent chapters are used throughout the rest of the book. In this part of the book we shall see that both main branches of topology meet at. one of the highest points common to both: in the AlcxanderPontryagin duality, which is the culmination of Part Four (see Chapter X IV )” The common parts of both branches (manifolds and bicompact spaces) may also be investigated further by means of the homology rings of locally bicompact spaces. Unfortunately, this subject is outside the scope of the book. 1

2

IIOMOLOdlC'AL MAN IFOLDS (/i-MAXIFOFDs)

The homology theory of compacta, one of whose main adornments is the Alexander-Pontryagin duality, can be developed in two ways. One possi­ bility is to transfer to compacta the fundamental notions of A-cycle and A-homology and to use these to define the A-groups of compacta. We used this method in Chapters XI and XII; the full development of this method leads unavoidably to topological homology groups and to the use of bicompact topological groups as coefficient domains (see Pontryagin [a]* and Steenrod fa]). The second method is to generalize the V-theory to compacta and bicompacta. This method was first proposed by A. X. Kolmogorov. The variant of this theory which is developed in the book is closely related to the V-theory of ordinary complexes. In the case of topological spaces the decisive advantage of the V-theory over the A-theory is that the former does not require the use of topological groups. Although it appears that the V-theory is more suitable to the investigation of figures more general than polyhedra, both theories are essentially equivalent, since they are dual to each other in the sense of the Pontryagin theory of characters. Chapter XIV is devoted to the introduction of the fundamental concepts of the V-theory and to the proof of the Alexander-Pontryagin duality. This proof depends on the combinatorial form of the duality (which is a theorem on subcomplexes imbedded in a combinatorial /(-manifold) treated in Chapter XIII. The elements of the theory of /(-manifolds is given in Chapter XIII. The first definition of /(-manifolds given in that chapter is an invariant definition based on the concepts of Chapter XII. But this dependence is only apparent, because only a few lines later we introduce the concept of a combinatorial h-manifold (a notion independent of Chapter XII) and it is this concept alone which is used in the remainder of Chap­ ter XIII. For this reason, Chapters XIII and XIV can be read without first reading Chapters XI and XII. In that case, however, XIV, 3.3 should be omitted. The concept of /(-manifold (introduced in 1926-1927 at approximately the same time by Alexander, Pontryagin and van Kampen) is the most natural generalization of manifolds for the purposes of homology theory: the definition singles out the local homology properties of ordinary mani­ folds—the homological homogeneity and simplicity of /(-manifolds is more accessible to investigation than topological homogeneity in the large. In addition to the elementary theory of /(-manifolds and the combinatorial form of the Alexander duality, Chapter XIII treats the Poincare duality and the stronger form given it by Veblen. In Chapter XV we deal with the more elementary form of the Alexaiuler* See the references listed at the end of the hook.

IIOMOLOO1CAL MAXll'OLRS (//-MAXIKOLbs)

3

Pontryagin duality, the duality an applied to curved polyhedra (in particu­ lar, the Jordan-Brouwer theorem). The exposition here depends only on Chapter XIII and on the simplest notions of Chapter XI; it is completely independent of Chapter XIV. The simpler form of the duality, which we call the “little” duality, is preceded by the theory of linking in Euclidean n-space. This theory belongs to the classical elementary parts of topology and should have its own interest for the reader. As references for the questions discussed in Part Four we may cite, in addition to the paper of Pontrvagin alluded to above: Aleksandrov [f (see vol. 1, Bibliography): k; 1], Kolmogorov [a; b; e; d], Alexander [b], Steenrod [a]. These papers contain references to the earlier literature.

Chapter X I I I H O M O LO G ICAL M A N IF O L D S (A -M A N IF O L D S ) §1. Definition and simplest properties §1.1 Definition of /i-manifolds. D efinition 1.1. An n-dimcnsional homological manifold, or simply an n-dimcnsional h-manifold, is a connected n-dimcnsional polyhedron whose local integral Betti groups in each point are isomorphic to the local Belli groups of Euclidean n-space. In other words, an a-dimensional A-manifold is a connected n-dimensional polyhedron 4? with the property that for every p £ the group ,/) is infinite cyclic, while A/(, J) = 0 (the null group) for every r < n. R e m a r k 1. We shall prove in Remark 3 that if 4> is an «-dimensional /i-manifold, then A/(, ?(), r < n, is the null group and Ap"(d>, ?[) is isomorphic to 91 for an arbitrary coefficient domain ?[. Theorem 7.2 of XII implies 1.11. If is an n-dimcnsional /i-manifold, then every triangulation K a of 4> has the following property: 1.111. If 0 a = 0 KaT a , T a 6 IIa , is oji arbitrary star, the group Aon(Oa) is infinite cyclic and Ad (Oa) = 0, r < n. R emark 2. Hence it follows from XII, Theorem 7.2, Corollary 5 that every (n — I)-simplex of I \ a is incident with precisely two n-siinplexes. Moreo\'er, I \ a is clearly a pure complex (AT, Def. 5.24). If a triangulation of a polyhedron has property 1.111, so docs every triangulation of ; hence 4> is an n-dimensional /i-manifold. Thus, the n-dimensional simplicial complexes satisfying 1.111, and only such complexes, are isomor­ phic to triangulations of /i-manifolds; these complexes are called com­ binatorial h-manifolds. In this chapter wc shall have to do only with combinatorial /i-manifolds and, therefore, the discussion will be essentially independent of XII. The invariance of the /i-manifolds, however, follows from the results of XII. R emark 3. A consequence of IX, Theorem 4.1 is the following proposi­ tion: If A0'(0„) = 0, r < n, and An"(Oa) is infinite cyclic, then Ar(Oa , ?() : 0 for r < n and A"(0„ , ?I) is isomorphic to ?(. This implies the asser­ tion in Remark 1. R emark I. An immediate consequence of Def. 1.1 and XII, Def. 0.2 is that every n-manifold (see 1, 5.3) is an //-manifold. §1.2. Elementary properties of //-manifolds. 1.21. If K is an n-dimnisional combinatorial h-manifold and c is a vertex -1

§1]

DEFINITION' AND SIMPLEST PROPERTIES

5

°f the star 0 Ke ?s an n-dimcnsional orientablc pscudomanifold (see YTII, Def. 3.11 and XI, LSI). In fact, since I\ is a pure /(-complex, it is easy to see that 0 Ke is also a pure //-complex. II T n 1 is an (// — 1)-simplex' of 0 Ac, 0 Kc contains the two /z-simplexes incident to T n~l in K. We shall now show that 0 Kc is a strongly connected complex. If this is not the case, then 0 Kc contains two /z-simplexes 7\" and 7V+/' which can­ not be joined by a chain of /z-simplexes. Denote by T f 1, • • • , Tf' those /z-simplexes of 0 Ke which can be connected with 7\" by chains of n-simplexes and let T’M 4in, • • • , 7’Mf/' be the remaining /z-simplexes of 0 Kc. Let K' be the complex consisting of the first kind of /z-simplexes and those of their faces which belong to 0 Ke and let K" be the similarly defined complex for the second kind of /z-simplexes. Then K' and K" are closed in 0 Kc, 0 Kc = K' u K", and dim (IC n K") < n — 2. Since e is a vertex of all the elements of 0 Kc, and hence is contained in every nonempty closed subcomplex of 0 Kc, it follows that K' n K" is nonempty. According to the definition of K' and K" it follows further that if T',~i is an (n — 1) -simplex of either complex, then both n-simplexes incident with T n~l belong to the same complex, K' or K", which contains T n~x. Therefore, assigning to e\rery //-simplex of K' the coefficient 1, we obtain an n-cycle (mod 2): -V G Z,"(K'; ./,) Q Z S ( 0 Ke, ./o). In the same way, the analogous //-chain in K" is a cycle (mod 2): G

./2) C ZT'(0A-e, ./,).

Since the cycles zin and zdl are not zero and are both in Z±n( 0 Kc, .Id), the order of Z±n( 0 Kc, J->) = A"(0Kc, ./••) is greater than 2. Since A„"(Oac) is infinite cyclic and A,i"'1(Oac) = 0, it follows that (IX, dffieorem 4.1) A"(0Kc, J-i ) is of order 2. This contradiction shows that 0 Kc is strongly connected; hence it is an /z-dimensional pseudomanifold. Since A0"(Oac) ^ 0, ()Kc is an orientablc pseudomanifold. This proves 1. 21.

Theorem 1.22. .1/z h-manifohl is a dosed psendomanifold. Proof. Let K be a triangulation of an //-dimensional //-manifold. In I'iew of 1.1, Remark 2, we need merely show that K is a strongly connected complex. Suppose that this is not so. Then I\ is a union of two nonempty pure n-dimensional closed subcomplexes K' and K" whose intersection is a complex of dimension < // — 2. Let c be a vertex ol K' n K". Then 0 Kc = ( 0 Ac n K') u ( 0 A-c n K").

(>

HOMOLOGICAL MANIFOLDS (/l-MANIFOLDS)

[CH. XIII

Since e c K' n K" and K', K" arc pure complexes, it follows that 0 Ke n K' and 0 Kc n K" are nonempty pure complexes. They arc closed in 0 Ke since K', K" are unrestricted complexes. Finally, ( 0 Kc n A") n ( 0 Ke n A") C K' o K"; hence the dimension of (0 Kfn IP) n ( 0 Kc n K") does not exceed n — 2. Consequently no n-simplex of 0 Ke n IP can be connected to an n-simplex of 0 Kc n IP' by a chain of u-simplexes. Therefore, 0 Kc is not strongly con­ nected and so it cannot be a pscudomanifold. This contradicts Theorem 1.21 and proves Theorem 1.22. Theorem 1.22 justifies D e f i n i t i o n ” 1.23. An /i-manifold is said to be nrientablc if it is an oricntablc pscudomanifold. In the contrary case it is called nonorientable. 1.24. In the notation of 1.21 the boundary BKe of the star 0 Kc is an (n — 1)dimensional simple h-manifold for n > 2. [A pseudomanifold (in particular, an /i-manifold) of dimension n is said to be simple if it is nrientablc and its A’-groups are null groups for 0 < r < /?.] Proof. Bv VIII, Corollary to Theorem 1.55, A0'+1(Oa'c) is isomorphic to Ao(BKe) for r > 0. This fact and 1.111 imply that A 4; for each of these values of n there are closed //-manifolds which are not manifolds. The fact is that for arbitrary n > 4 there are closed (// — 1)-manifolds (the so called Poincare spaces) which are not homeomorphic to the (// — ] ) -sphere but whose A-groups, in spite of this, are isomorphic to the A-groups of the (// — 1)-sphere. Assuming that this fact has been proved, we take in IE'1 1 a polyhedral complex K, which is a triangulation of an (// — 1)dimensional Poincare space; we assume also that all the vortices of K are in general position. Choose two points c0 and ci in II2"+l so that the set consisting of these two points and all the vertices of K is in general posi­ tion. If each point of I\ is connected to both points co , ci by straight line segments, we obtain two pyramids with common base I\ and vertices c0, . The union of the two pyramids is an /(-complex K "; it is easy to see that K" is a triangulation of an //-dimensional //-manifold. However, K" is not a manifold, since 0 KcQand 0 Ke( are not homeomorphic to the in­ terior of a solid //-sphere (an open n-cell). Hence the construction of //-manifolds which are not manifolds is re­ duced to the construction of Poincare spaces. The following example, taken from Seifert-Thrclfall (see vol. 1, Bibliog­ raphy, [S-T; p. 210]), is apparently the simplest example of a three-dimen­ sional Poincare space. The required manifold is obtained by identifying opposite faces of a dodecahedron, first twisting one of each pair of opposite faces through an angle t / o relative to the other. The exact scheme ol this identification is shown in Fig. 148, which shows 11 faces of the dodeca­ hedron; the twelfth face is the exterior of the figure. The identification of the elements (vertices, edges, and faces) is indicated in the figure by the assignment of the same letters or numbers to the elements which are to be

8

HOMOLOGICAL MAXIFOLDS (/l-MAXIFOLDS)

[CH. XIII

identified. In particular, the twelfth face is denoted by I. For a proof of the fact that the result of the identification is indeed a three-dimensional Poincare space the reader is referred to Seifert-Threlfall ([S-T; p. 21G ff.]). §1.4. Barycentric stars and /r-manifolds. Let K be an ^-dimensional h-manifold, and let K { be the barycentric subdivision of the complex K. The following proposition is of fundamental importance in the theory of //,-manifolds: 1.4. All the barycentric stars of the complex K arc simple pseudomanifolds. Proof. The theorem is obvious for n = 1; assuming that it is true for n — 1, we prove it for n. To this end, let e be an arbitrary vertex of I\ and assign to the centroid of each simplex Tr 0 Ke, r > 0, the centroid of the face T ~1 of TTopposite the vertex e. It is easy to see that this assignment is a (1— l) simplicial mapping (iso­ morphism) *S' of the outer boundary B* of the barycentric star T*"(e) dual to e onto the barycentric subdivision (BKe)i of the complex BKe (Fig. IT)). This mapping assigns to every simplex of B* with first vertex the centroid of some Tr = cTr~1 f 0 Ke, r > 0, the simplex of (BKc)i with first vertex the centroid of Tr~1 £ BKe. Hence, it follows easily that every barycentric star of K contained in B* (i.e., every element of the baryceutric complex dual to the triangulation K which is a subcomplex of the complex B*) is isomorphic to a barycentric star of BKe (that is, to a barycentric, star which is an element of the barycentric complex dual to the triangulation BKc). But B kc, by 1.24, is an (n — 1)-dimensional //.-manifold. Hence, by the

§ 2]

BARYCENTRIC COMPLEX OF A COMBINATORIAL //-MANIFOLD

9

inductive hypothesis, all the barycentric stars of BKe are simple pseudomanifolds; consequently, all the barycentric stars of K on B* are simple pseudomanifolds. Since every barycentric star of A of dimension < n — 1 is on the outer boundaiy B* of some //-dimensional barycentric star T*n(e), we have proved that every bciryeentric star of A of dimension < n — 1 is a simple pseudo-manifold. We need merely prove now that every //-dimensional barycentric star 77*”(e), i.e., every star dual to a vertex c of K, is a simple pseudomanifold. But this is an immediate consequence of 1.24 and of VIII, Corollary to Theorem 1.55 (also of 1.11 and the fact that T*"(e) is the star of e in the complex /Ci). This completes the proof of Theorem 1.4. R e m a r k . We have proved incidentally that the complex B* | the outer boundary of the barycentric star T*"(e)\ is isomorphic to (BKe)i (the barycentric subdivision of the outer boundary BKe of the star 0 Ke). §2. The barycentric complex of a combinatorial //-manifold §2.1. Notation; preliminary basic facts. Let K be a triangulation of an //-dimensional //-manifold !| K ||. We denote by AR the barycentric subdi­ vision of K and by T f the p-simplexes of /C; *ri \ f (p + q = n) will stand for the barycentric; stars dual to the simplexes T f. According to 1.4 and IV, 5.24, the baryeentrie stars of K form a fundamental system of subeomplexcs of th,c complex Ad (X, Def. 2.21). The same theorems imply a stronger form of this result, which we formulate below. Let K* be the complex of all the barycentric stars of K and let A,,* be a closed ///.-dimensional (m < n) subcomplex of I\*. Denote by XU1 C K l the barycentric subdivision of An* (see IV, 5.4), that is, the union of all the barycentric stars (as subcomplexes of AR) which are elements of Ad,*;

10

HOMOLOGICAL MANIFOLDS (/?■-MAXIFOLDS)

[CH. XIII

we know (IV, o.4) that 7v01 is a triangulation. With these conditions, IV, Theorem 5.24 and Theorem 1.4. of this chapter imply that the elements of AV form a fundamental system of subcomplexes of the complex Km . From this and X, Theorem 2.42 it follows further that: 2.10. If *fi,r are arbitrary orientations of the stars * l \ f £ /Co*, then the *t\l form a fundamental system of chains (X, Def. 2.41) of the complex K m •' If K 0* = K*, this theorem becomes 2.1. The system of all oriented baryccntrie stars *tif of the complex K is a fundamental system of chains of the complex 7T . We formulate separately the propositions expressing the fact that the fundamental system of chains *tu satisfies conditions 4° and 4° of X, Def. 2.41: 2.11. Every homology class 2 e Ar(Xoi) contains the barycentric subdivision Sk* of a cycle z* = 22 a * t u . 2.12. If a cycle zf = Si2*, where z* = 22 ?s homologous to zero in Ki , it is the boundary of a chain *.rr M = 22 fi*tuT+l§2.2. The complex ft*. We shall now construct, by the rule of X, 2.4, the (i-complex ft*, defined by the fundamental system of chains *fi/. The elements of the complex ft* will be denoted by ± * tf, where *tf corresponds to *h/ and —*tf to —*fi/. Since the pair of chains *fi/ and —*tu‘ is completely defined by the orientable pseudomanifold *TUT (as the pair of its orientations), the a-complex ft* is completely defined by the complex 7v* in the same sense that a simplicial complex 7v0 defines the (7-complex of its oriented simplexes. Hence, extending this analogy with simplicial complexes, we will denote the groups 7/(ft*),

^ (ft* ),

Ar(ft *),

V’(ft*), etc.

7/(7v*),

Z\K*),

Ar(7v*),

Vr(7v*), etc.,

by

respectively. We may now state the following fundamental result on the basis of X. Theorem 2.5: Tiikoukm 2.2. The groups A'(7v),

Ar(7vi),

Ar(7v*)

Vr(7w),

W(7v*)

are isomorphic-, the groups V’(7y'), arc also isomorphic.

§3 ]

THE INTERSECTION' NUMBER. THE POINCARE DUALITY

11

§3. The intersection number, the isomorphism IT and the Poincare duality §3.1. The intersection number X * 7 ). Let || I\ || be an oricntable n-dimcnsional /i.-manifold and choose a definite orientation of the com­ binatorial /t-manifold A, that is, orientations fi", • • • , of all its a-simplexes such that X! 7 is a evcle. Denote the induced orientations of the a-simplexes of AT by fi/1. We consider an oriented simplex L,}> of 1\ and the baryeentric star *l\i (p + (/ = n) dual to the simplex | t f |, and give this star an arbitrary orientation The assignment of the orientation *fi/' is equivalent to the assignment of the element *7 of the cell complex A*. Let | | be a simplex of AT contained in | 7 ’ j: | h j ’ | = (Rii(O)R-ii(i) ' ‘ ‘ Ci/tr-i)0!') > where alim < «i,u) < < ih, (see IV, o.l, where the vertices are writ­ ten in the reverse order'), i.e., ml(nj , auai , • • • , (h, are the centroids of the faces T

, < Tu\,) < •

< 7’,-f^n"-1 < 7 7

of the simplex 7 7 (Fig. 140). On the other hand, let ! fi/ | = (CnUli'U-t 1) ■’ ■ ali(n)) be a ^-simplex of *7’7 , with Ou

O

flij(jifl)

S-i

^

i.e., fli, , Uiop+i) , • • • , «u

iu /'-.r1) = ( - i ) V . Hence c' = —e. Analogously, if the orientations Lx = 7 I Qiro) ' ' ' Qu(k) • • • RuCi-iion-i) ■• ■ and 7/ 1 OX’

r\ —

7

I Rli'(0)

I • • •

• • • ttli(n) 1

are equal to the orientation L,,,/' and are therefore coherent, and , n—1 °1R0)

- - • G i / ( A - 1)01, -(/, -+!)

• • • aiiO it(p+l)

‘ ‘ ' O l i ( » ) I,

then (LxB: 0

= - ( j ,," : /" - 1),

(hx":;"-1) = ( - 1)%, = (-i)V , 7' = -7 .

Therefore, c'777' = ( —e ) v ( — 7) = £177,

so that £777 does not depend on the replacement of | t i / | by | t ^ j. In exactly the same way, using the strong connectedness of the complex *Tuq we prove that crjy is independent of the choice of | t\vq |. D efinition 3.1. The number c-qy = ± 1 , depending only on t /, *Uq and the prescribed orientation of the /i-manifold K is called the intersection number of t* and * tq in the oriented /i-manifold K* and is denoted by (t,p X *t q). R emark 1. Let Rn, /L /, Rn be the carrying planes of the simplexes | fix'' I, | h / I, | tn |, respectively, oriented in the same way as hx“, b /, tuq. Then X *t q) is precisely the intersection number ( /L / X Ru ) defined in A' 11, 2.1.

14

HOMOLOGICAL MANIFOLDS (/i-MANIFOLDS)

[OIL XIII

The intersection number (/," X *lf) is, by definition, set equal to zero for h X i (that is, if the barycentric star *Tn,'1 is not dual to the simplex Tf ) . Remark 2. If t f (*tf) is replaced by —t f ( —*tf), f ( 77) changes to —c ( —y). Hence ( ( - i f ) x %") = ( i f x ( - * t f ) ) = - ( t f x V ) . Furthermore, if the given orientation of the /i-manifold K is replaced by its opposite, y changes to —y. Hence the intersection number of t f and *// changes sign if the orientation of the /i-manifold is replaced by its negative. §3.2. The intersection number and the incidence numbers. Lemma 3.21. I f \ t f ~ l \ < \ t f |, then (3.2)

( t f X *//') = ( t f : t r ' ) ( * t f +1: * t f ) ( - i y ( t f - 1 X *t f +1).

Proof. Since | t f 1 | < I t f I implies that | *tf | < | * / / +1 [ and hence that ( t f : t f ~ l) = ± 1, ( * t f +l\*tiq) = ± 1, it follows that the right and left sides of (3.2) are both equal to 1 in absolute value. We need only show that both sides have the same sign. We consider first Ihe very important special case (3.21)

( t f i t f - ' ) = 1,

(7 /'+1:* 7 ) = 1.

Let ! h f

|

=

(flli(O )

' ' '

be a simplex on (contained in) | t f ], with ai,(0) , • • • , , fl1( the centroids of T m ° < ••• < T i(p- 2f ~ z < T f ~1 < T f (Fig. 141); and sup­ pose '/+i — (oijOiloil(lt+1) ••• 0i,-(„)) c *7 i f ; /1

that is, tlrq is coherent with the orientation *tu of the psevdomanifoJd *7Y,h Finally, suppose that the prescribed orientation of the pseudomanifold Ah is defined by the oriented simplex , H I I ox

— y | Qii(o) • • •

■■■ a u ( n) \.

To summarize: the orientation ,

p

> I

I

hfi — £ 1^T?(0) **’ Uli(p—2)(hjfll% I is coherent with t/; the orientation if/ __

hv

I

V} \ Cl\fCl l / ( p-}- l )

* * * 0-\i(n)

I

\

is coherent with *h//; the orientation .71 _

ox

I

|

— 7 I fln'(o) - - - Oit(p- 2)aijaiiaH(p +n • • • a u o o \

is coherent with the orientation of Ah. Therefore, for a given orientation of Ah (3.221)

( //’ X V ) = cm-

On the other hand, the orientation ( —1)hip7,~1 = ( —1)ve \ al7-(0) • • • o1;is coherent with t/'~l, the oi'ientation h /'”’ = f] \ aijauau(P+i) ■■■ aU(,o is coherent with *h/,rl and, as before, the orientation 7

| d i m

’ ’ *

2 )^ 1 j t t l i ( p + 1)

* * * ^\i(n)

|

is coherent with the orientation of 7vh . Then, for the given orientation of Ah , (3.23)

O r ’ X *//'+1) = ( — 1 ) Pcrjy,

i.e., (Up X V ) = ( - l ) f ((;H X *?/+1). Hence (3.2) is proved on the assumption that (3.21) holds. Assume now that (3.21) does not hold and let (/," :// ') = n j ,

( * //+1: V ) = fta ■

Then M r u r 1) = a-/ = i;

( v r r iM )

fin =

§3 ]

17

THE INTERSECTION NUMBER. THE POINCARE DUALITY

and, consequently, in view of what we have shown,

x % “)

=

(-iy \ t r i x

yji*tji+l').

But ((>' x % q) = r j ( t p X *t q) = ciJ(eiiiip X % q), u r l

x

= v jiitr 1x

Therefore,

(tip x *//') =

x

* t n = n j i - i y u r 1 x vj,*i;i+1) =

x v +l).

This completes the proof of Lemma 3.21. §3.3. The isomorphism IT and the Poincare duality. Let t,1’ be an arbi­ trary but fixed orientation of the simplex 7’/' of an orientable (and ori­ ented) n-dimensional /i-manifold K. Of the two possible orientations of the orientable pseudomanifold *7V/ denote by *q/' the one for which (tip X % q) = 1. We make this assumption for each dimension p and examine the resulting incidence matrices || *t q) || = || r?y/; |h First, if | / / ’-1 | is not a face of | i f |, then ‘) = 0 and ( *l q+l\ *t q) = 0. If l / ’~l | < j t f |, i.c., if { t r . t r ') X 0, we have by Lemma 3.21, {t,p X % q) = ( —I )p( l p:t/'~1) ( * l q+1:*tiq) ( t f ’~l X *tjq+l) ; or, recalling that (//' X *t q) = 1, ( / / ’ 1 X *tjq^1) = 1, 1 = ( - [ ) p( t r . t r l) ( * t q+l :*/,")• Therefore, multiplying both sides by ( —1)'’( / / : t/'~l) and remembering that ( / /' : / / ') = ± 1, we obtain (3.31)

( *t q+l:*t q) = ( —l ) l'(tiP:tjP~1).

We now assign to each chain xp f IT(I\, 21) the chain *xq £ Lq(K*, 21) which assumes on each * tq the same value that .r" assumes on t p\ (*xq • *t q) = (xp ■ tiP). We denote the chain */' by I)qxp. 'The mapping IT is obviously an iso­ morphism of LP( K) onto Lq(K*).

[CH. XIII

HOMOLOGICAL MAXIFOLDS (/l-MAXIFOLDS)

18

We shall prove the following fundamental property of Dq\ ADqxp = (° ‘o2)

V//'./•" = ( - l ) 7 r +1A:r'\

To prove this, we set J’-l _— \//)'./ /J-I'v) i f ij t% - t j

_V j i 1+1 _ (*4 9+1. •*/O,1\) \ 'j

for brevity and recall that by (8.81) 'kt-k A*tjQ+1 = ( - 1 ) " L , .ehi*th\ [The second equality follows from (8.81).] Therefore [remembering that ( t p x V ) = i], ( t f X A*/./+1) = ( - 1 ) PE * ckj( t f X V ) = ( - 1 Y c ^ t f X *tf) = ( - 1 y e ij} (A/ / x * //'K) = E *

X * //+‘) = e u i . t r 1 x %"+1) = Cij.

Consequently, d f x a * //+i) =

x v +1),

which is what we were to prove. C orollary to (8.44). Let *xQbe a cycle: A*/' = 0. Then on replacing p by p + 1 and hence q + 1 by q in (8.44), the left side of this identity becomes zero, and (A.rp+1 X *.r") = 0. But an arbitrary cycle zp homologous to zero may be written as A.rp+1. Therefore, 8.451. If V e ZY( K*) , z11 e HY ( K) , then (z” X V) = 0. Analogously, 8.452. If z" e Z / ( K ) and V £ /// ( A * ) , then {z“ X V) = 0. For, then Azp = 0, *zQ = A*.r9+1 and (zf X V) = (z“ X AV+1) = ( - l ) f ( A / X V ' 1) = 0. We rewrite 8.451 and 8.452 as one proposition: 8.45. If at least one of the two cycles zp and *z‘‘ is homologous to zero, then (zp X V) = 0. An immediate consequence of 8.45 is: If z f ~ z f (in A ), *zf ~ *zf (in A”*), then ( z f X *zf) = ( z f X *z->9).

23

THE INTERSECTION NUMBER. THE POINCARE DUALITY

§3]

This makes it possible to define the intersection number of two homology classes f d Y (A ) and G A"(A'*) as Y X Y ) = Y X Y ), where zp 4

*z" k. Y are arbitrary. R emark . If tlie coefficient domain is ./, Theorem 3.1b may he strength­ ened. Suppose Y C Z''(A*, ./), zp 4 //''(A, J ). Then there is an integer e such that ez1' C A''(A, ./), and hence (ezp X Y ) = 0. Rut (ezp X *2") = c(zp X Y ) , so that (zp X Y ) = 0. Therefore, 3.40. If at least one of the cycles zp and Y' is weakly homologous to zero (in A or A*, respectively), then (zp X Y y) = 0. §3.5. Veblen’s theorem. For every (,/, N)-basis (see A'l 11, Def. 2.3b) z p, • • • , z p of a complex A there is a (./, i)t)-basis *2/', • • • , *zf of the complex A* such that ( z f X *2/) = 5ij

( i , j = 1,2, • • • , it),

icd/i. 7r = irp = 7T7c^na/ /a the pth and qth Betti numbers of the manifold. || A || . Proof. The proof is based on (3.41). Indeed, by IX, Theorem 3.33, there is a canonical basis P

•U

J ’ ’ ‘

V

1

P

//1

l>

^1

7 P

P

I) 3

>‘

i

! f o ( p —

P

j ' * ‘ j MT(p)

1)

P

,

^1 ; * * *

)

^1 J

3'

)

J

/> ff(p)

of the module LP( K) which includes the cycles 2V’, • • ■ , z Z ) we consider the cocycles (3.,52)

Y , ••• , 2 /

in the dual basis (3. b 1)

u f , • • ■,

dr(p— 1), _

))

f'l , ' ‘ ' , p

T l , ‘ ‘ ‘ , ,l'r(p) ,

1) , -

P

2l , • • • , 2T, ~

P

2/l , ' ' ‘ , /M p) •

Since the isomorphism A" maps H VP( K) onto 1IZ(K*), no nontrivial linear combination of the cycles (3.53)

,

D \ p, • • • , I f z Z

is homologous to zero in A*. In order to show that these cycles form a (./, W)-basis of A* it is enough to show that each cycle Y € ZZ( K*) differs from a linear combination of the cycles D"z,f by an element of //"(A*). We shall prove this. First, Y = l)"zp, where z" 4 Z Z( K) . Because of the fundamental properties (see IX, 3.5) of the basis (3.51), zp = J2 c Y / + J2 bji)jP + J2 chzhp\

24

HOMOLOGICAL MANIFOLDS (/i-MANIFOLDs)

[CH. XIII

hence (since IT is an isomorphism)

V = l T z p = £ a iD"aip + £ bjDqv f + £ c,4/)V . Since some integral multiple of the cocycle £ (LiUi' + £ bjv/ is cohomologous to zero in K, the same multiple of the cycle *vQ = £ a;IT uf + £ bjD"vjP is homologous to zero in A*, that is, *z‘ — £ ChDqZh = *vq G //''(A*). Finally,' by (3.41), we have (z,p X Dqzjp) = ( z f • z / ) = . This completes the proof of the theorem. C o r o l l a r y t o V e b l e x ’s th e o r e m . Suppose that z" G ZP(K), but zp (f f l p(K)] thru there is a cycle *z‘ G Z'\K*) such that (zp X *zq) X 0. In fact, if z p, • • • , z / is a (./, i)i)-basis of K, there exist numbers Ci , not all zero, such that / -

£ c £ ' 6 HP(K).

Suppose, e.g., that Ci X 0. We construct, in accordance with Veblen’s theorem, cycles *zi'1 G Zffi(K*),i = 1, • • • , tt, such that ( z f X *z/‘) = 5^ . Then by 3.40, (zp X V ') = £ C i W X V ) - C! X 0. E x e r c is e . Prove the analogues of Veblen’s theorem and its corollary for the coefficient domains , m a prime.

§4. The combinatorial form of the Alexander duality §4.1. Formulation of the theorem. In this section K will denote a combi­ natorial ?i-dimensional orientable /i-manifold, K 0 a closed subcomplex of K, K* the complex of the barycentric stars of K and K(up, Ul A) = (.T,, +1 X if1- 1), where .r',+1 G L p H(K) is un arbitrary chain such that Axv+1 = v v. Theorem 4.51 implies that u(;+1 G / / H(/v) and A.t7'+1 = up. Formula (3.44) implies that Hu p, uq- 1) = (xvH X Ay") = ( - l ) H ,(A.rH1 X >/) = ( - i r +v

x yq),

and the proof is complete. The definition of U(wp, itq *) implies further: 4.531. Let K q be a closed subcomplex of K, and let / \ 0* C K* be the complex consisting of the baryeenlric stars dual to the elements of 7v0 ; if up G H/ ( Ko ) , then n{np, u q~1) = 0 for an arbitrary cycle v?—1 G ZAq- \ K * \ K 0*) homologous to zero in K*. For, by hypothesis, up = AxoP+1, where T0P+1 G Lp+1(Kt).

30

HOMOLOGICAL MANIFOLDS (/l-MANIFOLDs)

[CH. XIII

Hence, !)(«", uq-1) = (.r0!>11 X O

= 0.

The following proposition may be proved in a similar way (by means of 4.52): 4.532. If u9~l £ HAq- 1( K * \ , AY*), then \>(vr, uq~l) = 0 for any cycle up £ Z av( K q) homologous to zero in K. Theorems 4.531 and 4.532 immediately imply 4.53. Let v p and u p be two cycles of /v0 , homologous to zero in K and homologous to each other in I\a ; if u f ' 1 and are two cycles of K* \ An*, homologous to zero in K* and homologous to each other in K* \ A 0*, then v(vip, u r 1) =

u . r l)

(see Fig. 140, where a* is the outer boundaiy of the left hand triangle, oriented by the indicated arrows, v 2p is the boundary of the little triangle, oriented as indicated, and U\~l — u2 ~l). Now let Z akp(K») be the subgroup of ZA\ K f ) consisting of all the cycles homologous to zero in K. Then for arbitrary vf £ ZAKP(A 0)/ HAp(7v0) , it3-1 £ *Zq~1/IIAq~1( K * \ K l)*) we set (4.54)

(u71, it2-1) = t)(up, «3-1),

where ur and u‘~l are arbitrary elements of the cosets id’, u3-1, respectively; according to 4.53, the number P(u7’, it3-1) is independent of the choice of the cycles v p £ lF and uq~l £ u3-1.

P=l, qr2 F i g . 14G

§4]

COMBINATORIAL FORM OF TIIE ALEXANDER DUALITY

37

In particular, if Ap(/\) = 0 and Vp+1(/y) = 0, then A" \ K * ) = 0, and it follows that Za/ ( 7 v 0) / / / / ( / v 0) = A'Y/vo), *Zq- 1/ I I aq- ]( K * \ K 0*) = A9-1( / y * \ A V !); then (4.54) gives the Unking number t)(up, it9-1) for arbitrary UP e Ap(/v0), R9”1 € A9~1( K * \ K 0*). §4.6. Theorem on linked systems of cycles. In this subsection 91 will stand for either the field of rational numbers or one of the fields J m, m a prime. We shall say that two systems of cycles 2lp, •••

z;p € z p( K0 ,ai)

and z r 1, • • • , z r \

z r 1 e z r \ K * \ a v , ?q

are linked if i f z f , z f -1) = (Fig. 14b). T heorem 4.61 (Pontryagin). For every system of lirh (see VII, 6.420) cycles (4.611)

2ip, ••• , z / ,

2i" € Z>‘(Ko, 31)

of A’o ft f« possible to construct a system of cycles

(4 .6 1 2 )

z r 1,

••• , z,9“\

- r 1 € Z q- \ K *

\

Ko*, 91)

of I\* x A'0* r7hWi. is linked xcilh the system (4.611). Proof. We extend the cycles (4.611) to a canonical basis (4.613)

a-i", ///, ZiP, • • • , z,v, u f , r f

of the module / / ( / y 0 , 9(). In the basis _ n

M.- ,

- /)

- 7 7 - 0 - 1 )

Vj , Z i ,

_ n

■■■ , z , , Xk , Vi

dual to (4.613) we consider the cocycles zf, • • • , zf, and construct the cycles AZ)V, • • • , A D V of K * \ K 0*. By (3.41), (zf X I f z f ) = { zr ■z f ) = hi ; hence, by (4.52) D(2lp,

a d % ”)

= ( - i y ,+1( z f x ir-zf) = ( - i ) f \ .

HOMOLOGICAL MANIFOLDS (/l-MANIFOLDS)

38

[CH. XIII

The cvclos z r l = r —1)P+1AD % p are the required cycles. 4.611. For every system of lirh integral cycles (4.611)/

zip , •••, z . p

of Ko it is possible to construct a system of integral cycles (4.612)/ of K* \

••• , i , H Ko* such that \>(zip, z r l) * o,

Hzi“, z r l) = 0

(i^j).

Proof. Consider the cycles (4.611)/ as cycles with rational coefficients and construct, in accordance with Theorem 4.61, a system of rational cycles Ziq~1, ■• • , z,'q~l linked with the system (4.611)/ . Then, denoting by c-i the common denominator of the fractions which appear as the co­ efficients of the cycles z / q~l and putting z f -1 = c,z/7-1, we obtain the required system (4.612)/. T h e o r e m 4.62 (Pontryagin). Suppose the integral cycles (4.621)

Zi, • • • , z /

form a (./, di)-basis of Ko ■Then it is possible to construct a system of integral cycles (4.622) of K* \ Ko* which form a ( J , T )-basis of K* \ K 0* and arc linked with the system (4.621). Proof. We note first that since the system (4.621) is a (J , 9i)-basis of K 0, it can be extended to a canonical basis (4.613)

xf\ y/ , zu, ukr, v f

of 1 / ( K 0, 2f) (see IX, Theorem 3.33). Then, as in the proof of Theorem 4.61, we consider in the dual basis (4.623)

uf , v / , Z)f, xkp, y f

the cocycles -

V

-

P

zi , • • • , z / and construct the cycles (4.624)

A /)V , • • ■, ADqz / .

In exactly the same way as in the proof of Theorem 4.61 wc sec that

39

COMBINATORIAL FORM OF TIIK ALEXANDER DUALITY

§4]

the cycles (4.624), each multiplied by ( —1),+1, form a system linked with the system (4.621). The system (4.624) consists of ir = ?r,-1( K* \ K q*) cycles; therefore, to show that these cycles form a (./, 9i)-basis of K* \ K Q* it is enough to prove that each cycle zq~l C Z"~l( K * \ K 0*) differs from a linear combi­ nation of the cycles ADqzhp by an element of Hq~1( K * \ K0*). We shall prove this. First, by (4.4-10), (-1.626)

zq~l = A 7 )V +

where z0p e ZrP( K 0),

\

A'o*) c

\

K q*).

By the fundamental properties of the basis (4.623), Zov = X a{UiP + X M / + X hence, since A/)13 is a homomorphism of Z^P( K0) into Z^ \ K * \ K 0*),

A/)V = X a A l f u ? + X bjADqvjp + X ^ ^ V . Sinee some integral multiple of the cocycle v0p = X

+ X M /

is homologous to zero in K q and is consequently contained in CV(/v0), it follows by 4.420 that the same integral multiple of the cycle tT 1 = A D V = X fl/ADW + X M f l V is homologous to zero in K* \ A'0*, i.c., vq~l 6 n q~l(K* \

K q*) ;

[GIL XIII

HOMOLOGICAL MANIFOLDS (/l-MANIFOLDS)

40

hence zq~l = uq~l + vq~l £ Hq \ K * \ K 0*)- This com­ pletes the proof. In conclusion, we give an example illustrating the basic construction used in the proof of Theorems 4.01 and 4.G2. The construction consisted in forming the dual basis of a canonical basis of K 0 Cl K and operating on the former with the; operator AD'1. E x a m p l e . Let K be a triangulation of the 8-sphere *S'3; we think of the 8-sphere as the usual three-dimensional space compaetified b}" a single point at infinity. Let, the complex K q C K consist of the three sides and vertices of the triangle (ciC-ioO (Fig. 147). We take a canonical basis of the module L q ( K0) (see IX, 8.4, .Example 1

): 1 ,1 y1 — h

—

1 I I C2C3 I)

Vi

1

,1

— t‘>

—

1 I I C3C1 I,

Zi

I

. 1

= ty

, +

. 1 , b +

, 1

u ,

where tz — ) C]C2 |. In the dual basis (IX, 8.5. Examples of V-bases) - i

.1

V\ — £1

,1

£3 >

-1

,1

V a, y > /3, sueh that Sy “Za ^ Sy^Zfi

ill K y ,

i.e., if Sy“Za — Sy^Zp £ Hy (Ky). It is easily seen that the relation of cohomology between cocycles of the complexes of a spectrum X is reflexive, symmetric and transitive. We give a short proof of transitivity, the only property which perhaps requires it. Suppose £ ZvT( K a),

z0 £ ZVT( K0),

zy £ Z vr( K y)

and S\aza

S f z 0 in K\ ,

A > a,

Sfzg ^ >%yzy in

A > /3,

m > /3,

,

ix > y.

Choose K„ so that v > A, v > ix. Then s ; s xaza Since

".QA.. SSSfzp

SfsSzff,

SSS/zy

in K

S f S / z 0 in A'„ by (1.13), it follows that • S,xSiaza -

S : S , yzy in K , ,

i.e ., Za

^

^y

m ,TC.

Therefore, the cohomology relation partitions the set of all r-cocycles of all the complexes K a of a spectrum ."hi into r-dimensional cohomology classes of JC (further indices will be attached to whenever necessary). We shall now define addition for the cohomology classes. We note first that two arbitrary cohomology classes frr, contain elements z j and za" belonging to the same complex I \ a £ JC: z f £ Z / ( K a),

za" £ Z vr( K a).

In fact, choose arbitrary za, £ fi , let. I£a £ IK be such that a > «i , a > Z j = S aaiZa, £ f / ,

za, £ lif, and set Za" = S aa’Zat £ f / .

To define the sum of two cohomology classes and choose arbitrary cocycles z j £ ^ and za" £ f 2r of the same K a ; the cohomology class

§

1]

COHOMOLOGY GKOGPS OF BICOMPACTA.

43

containing the cocycle za' + za" is, by definition, the sum of and f2r. Il is easily verified that this sum depends only on and f 2r and not on the choice of z j 6 f i , za" 6 f /. To show this, suppose V e r;,

ZfT e f2r,

e V (^ )>

V' e V ( ^ ) -

We must show that S y ‘ (za' + Za") -

+ V')

in some K y , y > a, y > (3. Since z j ^ z j in X, there exists a K yi , y { > a, 71 > j3 such that S yiaza' -

S y, W

in K yi ;

since zj' ^ z/' in X, there is a 7v72, y2 > a, y2 > (3, such that ■VW -

s„ v .

Take 7 > 71 , y > y2. Then S 77,). R em ark 1. All the arguments and definitions of 1.2 can be repeated using closed coverings a, 0, y, ■■ ■ of instead of open coverings of 4>; it can be proved that the cohomology groups defined by means of closed coverings are isomorphic to those defined by means of open coverings. R emark 2 . The definition of the cohomology groups given above is

formally applicable to arbitrary topological spaces; the cohomology groups defined by open and closed coverings an* also isomorphic in normal spaces. However, our definition can be considered as final only for bieompaeta; it is expedient to use another definition even in the case of locally Incompact normal spaces (sec Aleksandrov [k; 1; f (see vol. 1, Bibliography)]). §1.4. The group Vnn. Let K a be a finite unrestricted siniplicial complex. We denote by Z xw( K a) the subgroup of the group Z X°(K) consisting of all 0-cocyeles constant on K a (see XIII, ‘1.2, the case p = 0). Remark 1. The elements of Zv ( K a) are 0-cocycles extensible over an arbitrary unrestricted simplieial complex K Z5 K a . Now let I\ a and K g be elements of a spectrum X, and suppose that Za G Z v ° ( K a ), Zgn f Z x \ K g ) . We shall say (see Def. 1.15) that z a° and zg° are contained in the same V00-elass of the spectrum 3C if there is a 7, y > a, y > 0, such that

‘1G

V-GROUPS OF COMPACTA. ALEXANDER-PONTRYAGIN DUALITY

Cll. XIV

It is easily verified, us in 1.1, that this is a proper definition. With addition for the V^-cl asses defined in exactly the same way as for the V-classes, the set of V00-classes becomes a group which we call the V°°-group of the spec­ trum 3C and denote by V00(JC). If 3C is the full spectrum of a bicompactum , then we call the group V00(3C) the group V00(). R emark 2. We leave the proof of the following theorem to the reader: 1.41. If 21 is the coefficient domain and the bicompactum 4> has a finite number ir° of components, then V00(4>) is the direct sum of 7r° — 1 groups, each isomorphic to 21. The proof is based on X I11, 4.3 and on the fact that every K a £ 3C con­ sists of no more than t° components, while for every K a there is a Kg > K a with 7r° components. §2. Cofinal suborders of spectra. The case of compacta §2.1. Cofinal suborders of spectra. D efinition 2.11. A partially ordered set O' is said to be a suborder of a partially ordered set 0 if every element of O' is an element of 0 and if £1 £ O',

.t2 £ O',

£1 > £2 in O'

implies that £1 > .To in 0. A suborder 0' of a directed partially ordered set 0 is said to lie cojimd with 0 if 0' is a directed partially ordered set and if .r £ 0 implies that there exists a y £ 0' such that y > x

in 0.

D efinition 2.12. A spectrum 3Zf is said to lie cofinal with a spectrum .7C if the following conditions are satisfied:

1°. The spectrum ,TC', as a partially ordered set consisting of complexes, is cofinal with the partially ordered set ,1C. 2°. If K g > K a in K', then every projection of Kg into K a in K' is a pro­ jection in 5C. hvery spectrum JC' cofinal with the full projective spectrum JC of a bi­ compactum is called a projective spectrum of 4>. T heorem 2.13. If a spectrum JC' is cofinal with a spectrum 5C, then Vr(3C) ~ Vr(0C'). Proof. The set of all r-eoeycles z j of the complexes of 3C' is a subset of the set of r-cocycles of the complexes of 3C; since z j ^ z f in X' implies that ^ z f in ;fC, every V-class of JC' is contained in a (unique) V-class of X. It is not hard to see that the resulting natural mapping of V^.TC') into V’(3C) is a homomorphism.

COFINAL SUBORDERS OF SPECTRA. COMPACTA

47

We shall show thax this homomorphism is an isomorphism onto. To this end, it is sufficient to prove that every V-class of X contains precisely one V-class of X ' . Let f be a V-class of X and suppose za G f. Since X' is cofinal with X, there is a complex Kp G X' such that Kp > K a in X. Then the cocycle zp = Spaza of Kp G 3C' is contained in a V-class of ,X'; since has an element zp in common with it is contained in f. Hence every V-class of X contains at least one V-class of X ' . It remains to be shown that no V-class of 3\ contains more than one V-class of X ' . To this end, if suffices to prove that two cocycles 2a G Z v

( K a ) ,

Zp G

(Kp),

K a G X',

I\p G X',

cohomolugous in X are also cohomologous in X'. Suppose that 2a

zp in X.

Then there is a complex K yi G tK*, j i > «, yi > j3, puch that S yiaza

S yfzp in K yi .

Choose a complex I \ y„ in X' such that K yi > K y, in X; such a complex K G X' exists, since X' is cofinal with X. Xow choose a complex K y G X' such that K y > K a , Kp , K y, in X'. The complex K y exists since >TC' is a directed partially ordered set. Projections S ya, *§/ are defined in X ' ; since both mappings are at the same time projections in ,tCand since y > a, y > y > yi in ITC, it follows that ey £ ^ IQ J a~ fi Q , ^ C'y t'i ^C' "n , ^ c y 71fQ j y^ in I \ y , i.e., S yaza ^ S y zp i>i K y . This completes the proof. C o r o l l a r y . The VT -groups of a bicompactum arc isomorphic to the Vr-groups of an arbitrary spectrum of 4>. §2.2. The case of d? a compactum. T heorem 2.21. Let 0 be the partially ordered set of all open coverings (see I, 8.1) of an infinite compactum fI>; and let 0' be a suborder of 0 satisfying the following condition: for every e > 0, 0' contains an e-covering of dp. Then 0' is cofinal with 0. R emark 1. The converse is obvious: if 0' is cofinal with 0, than 0' con­ tains an e-covering of for every e > 0. Proof of Theorem 2.21. Let a be an arbitrary covering of and choose an «i > a, c*i G 0. There exists (sec T, Lemma 8.34) an e > 0 such that every

48

V-G[tOTTS OF COMI’ACTA. ALEXAXDER-POXTRYAGIX DUALITY [CH. XIV

e-covering of 4> is a refinement of ax and is therefore > a. If y £ O' is an arbitrary e-covering, we see that y > a. It remains to be proved that 0' is directed. If a, /? £ 0', choose a covering ai £ 0 greater than both of these and determine the corresponding e > 0 and y as above. R emark 2. As the partially ordered set 0' of Theorem 2.21 we can take an arbitrary sequence of coverings (2.2)

0' = {aa , o2 , • • • , ak , ■■• },

where ak is an et-covering and lint ek — 0. We may assume that ak+i is a refinement of ak ; hence we can suppose that O' is a countable ordered set of the same order type as the natural numbers. The spectrum SZ' corre­ sponding to the set 0' is cofinal with the full projective spectrum 3C of , and is therefore a spectrum of 4>. It is countable and has the same order type as the natural numbers. Next, if the sequence (2.2) is chosen so that for arbitrary /.• every element of ak+i is contained in precisely one element of ak (see Remark 3), we ob­ tain a spectrum .TC' of L which contains exactly one projection S ahah of ak into ah for each pair of coverings ah and ak , k > h. Finally, if dim = n, we can choose ?i-complexes as the elements of the spectrum JC'. It follows, in particular, that 2.22 If dim T = n,.and r > n, then V’’(. Consider the sequence of complexes N , 7\ i , 7v2, *

, 7v „, •

,

each of which is the barycentric subdivision of the preceding. Denote by av the open covering of whose elements are the open stars of the triangulation K v (see LV, I)ef. M.20). Then 7v„ is the nerve of , so that we may regard the notation K v as an abbreviation of K ar . Let ef H be an arbitrary vertex of the complex 7v„+1 , and denote by

C'OrrXAL SUBORDERS OF Sl’ECTllA. COMPACTA

49

' “ cf a V('rtcx of the carrier of the point, c / +1 in K, . The mapping = l| K a" |j d I’ and is mapped by a canonical displacement into a cycle z f d Z f \ K a"). This mapping induces a homomorphism S of the group Z'(T) into Z \ V ) , which maps 3e'(F) into //'(F ). The homomorphism S induces a homomorphism (denoted by the same letter) of A '(I1) into A'(F). The homomorphism S maps A'’(F) onto A'(F). Indeed, every homology class f (E A'(F) contains a cycle z f of a triangulation Qa" (see 3.21). Obviously, the homomorphism S maps the proper cycle ( z j , zai , • • • , zuf , ■■■), where zaf is the barycentric subdivision of z„' of order h, into z . It remains to be proved that the homomorphism S of A'(F) onto A'(F) is an isomorphism. To show this it is sufficient to prove 3.351. If z = Ss € //' (F), then 5r G 3Cr(F). To prove the last assertion, denote by Qan the complex consisting of all the simplexes of a sufficiently fine simplicial subdivision of ,S" whose closure is contained in r. We may assume that Z = US* =

€ IlAiQa"),

that j r is a proper cycle of the polyhedron Qan and that the displacement S is effected in this polyhedron. Then 3.351 can be proved in exactly the same way as XI, Theorem 4X1. §3.4. The Alexander-Pontryagin duality and its fundamental conse­ quences. The fundamental theorem of this chapter may be formulated as follows: The Alexaxdek-Poxtuyagix D u ality. Let 4> d Sn be a compaction, and let F = S n \ . If p and q are two nonnegative integers whose sum is n, then the groups W'() = ^""(T) for p = 0 and A,/-l(F) = A°°(F) [i.e., Am(Kr)}for p = n — 1. Ke.mauk. In particular, the number of components of F is 7r"~l() + 1, where 7r" '(T) is the rank of V'1 l(T). The Alexander-Pontryagin duality is one of the fundamental results of modern topology. Before we go on to prove it, we shall list some of its consequences and special cases. These will serve to illustrate its importance. First, because homcomorphic compacta 4> have isomorphic groups Vr(T), the Alexander-Pontryagin duality implies 3.41. / / and T are homcomorphic closed subsets of *S", the groups A'(,S7' ) and A' (S" X yF) are isomorphic. In the same way, the Alexander-Pontryagin duality and Theorem 3.31 imply

§3]

FO RM U LA TIO N

OF

Till',

A L E N A \ I) E R - l ’O N T R Y A G I X

DU A LITY

r>3

3.42. If F and I1' arc homcomorphie open subsets of I\ ", then the groups Fr( S n \ I') and Vr(*S" \ I1') are isomorphic. A special case of Theorem 3.41 is 3.410. Two homcomorphie closed subsets and 4>' of S n separate S ’1 into the same number of components. Hence (see II, Defs. 1.20 and 1.291), 3.40. If one of two homcomorphie closed subsets of Rn is an absolute (regular) boundary, then the other is also. X o\y suppose that P C S" is a curved polyhedron. Then (IX, Theorem 3-31) (3.440)

V0P(4>) = 0 P '(4>) + X /(T )

has a finite number of generators. 'Therefore, A,g '(F) also has a finite number of generators; consequently, (3.441)

V " ‘( r ) = 0 r'""‘( r) + A ),r'(r),

where Au0g-1(F) is a free group and 0 ,,-1(r ) is the finite subgroup of A04“‘(F) consisting of all the elements of finite order of Ao^’O’). The group 0"“‘(r ) is called the (q — l)st torsion group of the open set F. The isomorphism V0P(4>) A(X’(F) and (3.440), (3.441) imply that 0 P-1(T) « ©a-1(r ), A ,/(T ) ^ A ^ F ) . Hence, T h e A le x a n d e r D u a l it y F o r C u r v ed P o ly h e d iia In S":

If 4> d S n is a curved polyhedron and p + q = n, then the pth Betti number of 4> -is equal to the (q — l).q Betti number of the open set F = Bn \ )

for every curved polyhedron T. Therefore: T he A lexander D uality (mod m) (for curved polyhedra 4> c >S”). The groups A ,/(T ) and Amq !(F) are isomorphic. Finally, let 4> be an (n — 1)-dimensional closed pseudomanifold in S n. Then Ao71”1^ ) is of order 2, so that A2on(r ) is also of order 2, i.e., I’ con­ sists of two components. But then A(,°"(r) is infinite cyclic; hence A"~‘(T) is also infinite cyclic and therefore 4> is orientable.

54

V-G UO UPS

OF f'OMPAOTA.

AL EX A N 'O E R -PO XT R Y A G IX

DUALITY

ffJI.

XIV

Since An~\Kv) = 0 for an arbitrary closed proper subcomplex 7v0 of a triangulation A”"-1 of the pseudomanifold I1, no polyhedron of the form |( AY'”1 I!, and hence, as is easily seen, no proper closed subset $0 C separates A'1. I11 other words, is a regular boundary (see II, Theorem 1.2). Therefore, 0 .44. The Jordax-Brouweii Theorem. E v e r y (n — \ ) - d i m e n s i o n a l closed p s e u d o m a n i f o l d i n A" is orientable, sepa ra tes S n i n to p re c is e ly tw o d o m a i n s a n d i s the c o m m o n b o u n d a r y o f these two d o m a i n s .

Since no closed proper subset of a closed (n — 1)-dimensional pseudomanifold contained in A" separates A7', then in view of Theorem 3.410 the same is true of sets homeomorphic to proper subsets of such pseudomani­ folds. It follows that no interior point of a compaction 0 there is an t-ncighborhood of q whose boundary is homeomorphic to a proper subset of an (n — 1)-sphere. Since these characterizations are topologically invariant, it follows that: 3 .4 0 . A topological m a p p i n g o f a c o m p a c t u m 4> is finite. Suppose that consists of a finite number of points 0i , 0 2 , • • • , on . In view of the Remark at the end of 2.3, to prove the Alexander-Pontryagin duality in this special case it is necessary to prove the following propositions: R

to

e m a r k

.

1°. T h e g r o u p A n 1( P, ?() is the direct s u m o f tt — 1 g r o u p s is o m o r p h ic

?I. 2°. Ar(T, 51) = 0 for r < n - 1. Proof of 1°. Let A'i , K > , ■■■ , K h , ■■■ be a sequence of triangulations of A”, each of which is a subdivision of the preceding, and such that mesh Kh approaches zero with increasing h. With no loss in generality we may assume: 1) For every h each ot is contained in an //-simplex of K h , and no nsimplex of K h contains more than one of the points o{ . to

FO RM U LA TIO N

§3]

OF

THE

A L E X A X D E R - l ’O X T K Y A G I N

D UALITY

OO

2) The simplexes 7\i of K>t are indexed so that o, is contained in T'h for i < We choose a definite orientation of S n and agree that all the 7 V have the induced orientation. The result is a set of oriented simplexes jrbi, ) = 0,

that = A( — Xf>T hi"),

i.e., L r - i^ - O

in Qi> •

hi For arbitrary h, Zhi.

71—1

Zu

71—1

• 111 1 .

Indeed, a projection of the boundary of 7 V onto the boundary of I \ , n from oi yields a deformation of zhi l~l into Zu" 1-, hence the t A v o cycles are homotopic and therefore homologous in F. In A'ieAY of b), instead of 3.51 it is sufficient to prove 3.510. EA-ery cycle z n~l € Z n~l(Y, 31) is homologous in F to a cycle of the form X V w v 1 for some h depending on zn \ We shall proA’c 3.510. By 3.21, ^r”-1 is homologous in K h to a cycle zkn~l Ka4hin is a chain bounded by X a-j1 C;Zh i ’‘~ l , then ACHiTi1 c,4;” — E -> ' aAi") = 0. Since Kk is an a-dimensional pseudomanifold and X X * ~ Xo*- nQ.i" is an n-cycle, the coefficients of all the thin in this cycle must be the same. Since the coefficient of fo,” is zero, it follows that all the a (and all the a,) are also zero. This completes the proof. We denote by the element of An-1(F, 3f) containing the cycle It follows from 3.51 and 3.52 that every element of A7'-1(F, 31) is uniquely representable in the form ^n-i =

c . £ 3(

(f < 7T -

1),

i.e., that An-1(F, 31) is the direct sum of ir — 1 groups each isomorphic to 31. This proves 1°. To prove 2°, suppose that z 6 Z’’(r> 3f), r < n — 1, and that zr is a normal cycle for r = 0. Let o' be a point of F such that none of the planes spanned by o' and any of the simplexes z contains any of the points (a , • • • , m . The point o' always exists, since these planes have dimension < r + 1 < n - 1. Xow deform the vertices of z into o' along the straight lines joining o' to these vertices (the deformation may be pictured as a uniform flow of the vertices of z into o' which takes place in a unit, of time along these straight lines). This is then a deformation in F of z into the r-cycle identically equal to zero (i.e., all its coefficients are zero). Hence every r-cycle of F is, in our conditions, homotopic, and therefore homologous, to zero in F, which was to be proved. This completes the proof of the Alexander-Fontryagin duality for a finite compact um. H- The complexes K a , K„a , K a*, K Qa*, Qal; the sets Ga and Va (see IV, 5.4) §4.1. Definitions and notation (see Fig. 149). In the rest of this chapter, F is an infinite closed subset of A” and F = S" X T . Let A be a triangulation of S n. Let />• be a natural number; denote by A”*, the /Ah order barycentric sub-

§4]

THE

COMPLEXES

K a , A'0f[ , A’,*, AT»*, Qai

.

THE

SETS

(7„ , I1,

57

division of A, and by AT* the subcomplex of A* consisting of all (be simplexes of AT containing points of and all their faces. It is clear that (4-U )

|| A'oft || ,

(4-12)

l! A'o.m i I! C || K 0k ||

for all k. The body of the complex K k \ AT* will be denoted by (1* . Let AT* be the complex of the barycentric stars of AT , and let AT** be the subcomplex of AT* consisting of the barycentric stars of AT* dual to the simplexes of AT* . Since AT*- is a closed subcomplex of AT , AT** is an open and AT* \ AT** is a closed subcomplex of AT*. Therefore, the body of AT** is open in S ' and the body of A a* \ AT** is a polyhedron, one of whose triangulations is the barycentric subdivision Qkl (see IV, 5.-1) of AT* \ AT**. The set of all interior points relative to A" of the polyhedron || Qn || = I! AT* !| \ || AT** || will be denoted by IT . In view of IV, 5.-12, we have (-1.13)

T c || Ao* II C || A V |j ,

r 3 Qk 3 Gk 3 li Qkl || =3 IT .

Let e* be the mesh of AT (i.e., the maximum of the diameters of the simplexes of AT). Then T C || AT** |i C *S'(T, 2e*), (4.1-1) I’ 3 IT 3 S n \

*S'(«IJ, 2ek) ;

since lim e* = 0, (-1.15) Furthermore,

I1 = UL.i IT . since

|| AT* |]

C

|| AT** ||

and

|| AT** ||

is open,

p(T, || Qkl || ) > 0.

We now set ki = 1 and, assuming that ka has been defined for the natural number a, denote by the first natural number satisfying the following conditions: 1°. 2 £*a+1 < p(T, |KT„i!! ).

2°. 2e*o+1 is less than both e*o and the maximum of the diameters of the sets T n T a , where Ta £ K ka . R e m a r k . If 0 a is an open star of the triangulation K ka , we see that be­ cause of 2° it cannot happen that every one of the sets T n 0 a is contained in some T n 0 a+i (i.e. the covering ! consisting of all the sets T n 0 a+1 follows (in the sense of 1.3; see also I, 8.1) the covering

58

V -G U O U P S O F C'OMPACTA. A L E X A X D E I t - P O N T K Y A G lN D U A L I T Y

W Z& F

i g

Qk |

[C H . X IV

r ocl

. M l )

c( insist in”, of nil n (J„ . Condition 2° was introduced for precisely this reason. Condition 1° implies that

Si*,

c

s- \

: oM r ;

hence IV, M => 8 n \

S(*, 2e*a4I) =) ■; Ca„i II =5 IV, •

For convenience, we now introduce the following notation. The com­ plexes AC,. , AVa.„ , AW, AW*, Qka\ will be denoted by K a , K Oa , A. a , A.oV , Qai , respectively; the open sets IC„ and CA -n will be written as IV and Ga ■ Finally, the barycentric subdivisions of the complexes AC , AVa will be de­ noted by ACi , AV„i . Because of what we have already proved in this subsection, we have (4.10)

r = u w || Qal

= u:_! i v ,

§4]

'I'llE

C O M PLEX E.S

AT , A'„u AT*, I f f ' , Q„1

(4.17)

.

THE

SETS

G„ ,

a

r>n

0 a 3 || Qal || ,

(4-IS)

3 || Qal || 3 v „ .

§4.2. The continuous mapping C of G„ onto || Qal ||. We shall prove the following proposition: 4.21. There exists a continuous mapping C of Ga onto || Qal || , whose set of fixed points contains |! Qol j| . Proof. Let Ta/ be a simplex of ATi , which does not belong to Qai , Imf is contained in a simplex 7V° G If, \

If a•

Then 7 V = (7’„"11 >

> 7V"),

where Tan° G AT \ A'0a , but T f ‘r G Koa . Ltd T f l+' be the first of the simplexes T f ‘", • • • , T„,,r contained in K t]a . Since K nu is a closed complex, we have 7 V C

A'oa , • • • , T f H G I f a

and 7T/ = (7Y ‘° > 7TT = ( 7’a”’+1 >

> 7 Y ‘) G Qai , • * > Ta^) G A'o„l .

Every point p of the simplex T af uniquely determines the segment p"p' containing p and joining a point p" G T a\ with a point p' G 7Tm Set C(p) = p'■ The mapping C is defined for all p G Ga \ || Qai || . If p G |i Qai || , set C(p) = p. The mapping C is continuous and satisfies the re­ quirements of Theorem 4.21. §4.3. Some properties of the group Z \T ). It follows from (4.10) and (4.18) that every chain x of T is a chain of IT , and hence of |] Qai |] , for every sufficiently large a. Consequently, in view of 3.21, every cycle z r Zr( r ) is homologous to a cycle zai G Z'(Qal) in T. But Qai is the barycentric subdi\’ision of A„* \ Eoa* Cl AT*; hence, because of XIII, Theorem 2.1 1, zal G Z''(Qai) is homologous in Qni , and therefore in L, to a cycle of the form .s'uiTa*, where za* G Z±r(K„* \ AY,*). Therefore, 4.81. Every homology class f G W(L) contains, for every sufficiently large, a, a cycle, of the form saiaza*, where Za* G Z f ( K a* \ I f a * ) . Furthermore, XIII, Theorem 2.12 implies 4.32. If saf z a* G Ilf(Qai), 2a* G Z f { K * \ Za* G IlAT( K a * \ I f a * ) .

I f f ) , then

GO

V - 0 ROLFS OF COMPACT A . ALEXA X D ER - P O XT R YA G I X DUALITY

[('II. XIV

§5. A combinatorial lemma §5.1. Introductory remarks and formulation of the lemma. Since K ai is the barycentric subdivision of K a , every simplex' T ai A K a\ is contained in exactly one barycentric star of K a . We shall call this star the carrier of the simplex T al in AV ; a simplex T al A K ai is said to be a principal simplex if its dimension is the same as the dimension of its carrier in K a*. It is easily verified that a principal simplex T a\ A K a\ is of the form (5.11)

Tal = (7V ‘ > Ta’- 1 >

> Tap),

where T an A K a , • • • , T ap A K u are the simplexes whose centroids are the vertices of T a\ ■ To show this, suppose that *T q A A a* is the carrier of the principal simplex T a,9. The barycentric star *Taq is dual to a simplex T av A K a , and, by definition, the cy-simplexes of *TaQhave the form (5.11). If T p A K a and *Taq A K a* are dual, we shall say for convenience that the simplex T aP is dual to each of the principal simplexes T a\ A *Ta9. Let K a and Kp , /3 > a, he two of the complexes Ah , Ah , • • • , K a , (sec 4.2), and let A'/ be a normal simplicial mapping (see X, 3.1) of Kp into Ka . We denote by Spd" the subdivision operator in Kpi (i.e., the opera­ tor which maps the chains of any K a> or K „ Tp"> > ••• > 7 V r) 6 A'„ the sini])lex (5.21)

S j lTpXr = T u] = ( S f T / " > ■• • > * 7 7 7 " )

We now investigate the effect of f).22. AVer// (/-s im p le x

r

A'„,.

7 on the principal simplexes of Kp{ .

'V’7 = ( 7 7 11 > 7 «"* > ••• > Ta”V), which As not a principal simplex, satisfies the condition ntJ < />, For, since n, > 1, n > nn (0 < i < q = n - p), then nq > p would imply that n0 = //, /o = n — 1, • ■• , n,, = p, i.e., it would imply that 7’7 is a principal simplex. This proves 5.22. If we now ask which principal simplexes (5.23,)

Tpi = ( i Y >

> 7’/ ) 6 A7

are mapped by S a\01 onto a given principal simplex (5.23a)

T ai = ( 7 7 >

■■ > 7 7 ) G K a, ,

we find that if S j lTm9 = 7’7 , then ,SjV," = 7 7 . On the other hand, if S j T p n = 7 7 , then $ 7 ’ maps precisely one simplex of the form (5.23,) onto the given simplex (5.23„). In fact, if S ai lTgiq = 7’7 , where Tp q, T a q are of the form (5.23,) and ( o.23q), then each of the simplexes 7’,"-1, • • • , 7’, p is uniquely determined as the unique face of 71," mapped by S j onto 77~"\ • • • , 7 7 , respectively. Hence, the principal simplexes Tpiq £ A,i mapped by »S7' onto a given principal simplex (5.23„) correspond ( 1 - 1 ) to the simplexes 7’, “ A A', mapped by S j onto 7 7 R e m a r k . Denote b y S j the (nondegenerate) affine mapping of the closed simplex f p n onto T a" induced b y the mapping S j of the vertices of Tpn onto those of T aTi. Since the nondegeneratc affine mapping * 7 of the closed simplex f p n onto 7 7 maps the centroid of each face of Tp1into the centroid of its image under S j , then O /91m

> Ta"q)

be an arbitrary (/-simplex of 7vai ; if Taf is a principal simplex, then the orientation ta/ of Ta/ has already been prescribed; if Ta/ is a nonprincipal simplex, we denote by ta/ any orientation of Ta/ . Let Tguf = ( Tg. / > T g . r 1 >

> Tg. / )

(1 < j < v)

be all the principal simplexes of I\g\ mapped b}r S a/ Xonto T a/ (of course, distinct simplexes Tgi,/ may be dual to the same simplex Tg, p, so that the simplexes Tg,/’ are not necessarily distinct for distinct values of j). Let S a/ ltgi./ = v/aiQ, where 77, = ± 1 . Because of the definition of the operator S a/ Xand (5.25), we have (5.27)

(Sai^Vgi1 ■ ta/ ) =

(UgZ • v/ gu/ ) =

VjU'aP ■ S j t g , / ’).

We consider two cases: 1°. T a/ is not a principal simplex. Then by 5.22 and (5.20), nq < p; since S j T g f = Tan\ ( .r / • S a% f ) = 0 for ally = 1,2, • • • , r. In con­ junction with (5.27) this yields ( S j ' y g Z ■ laZ) = 0. In view of the definition of .sala, the value of the chain sa/ J ) aqxap on a nonprincipal simplex ta/ is also zero, so that (r iW

■ l*Z) = (salaDaq.r / • Ia/ )

for a nonprincipal T a/. 2°. Tai is a principal simplex. Then (5.20) becomes T*Z = ( Tan > TaH~x > • • > 7’/ ) ,

04

V - G K O U P S O F COM PA CT A. A L E X A N 'D E R - P O X T R Y A G IN D U A L I T Y

[CH. X IV

:uid $ r v

& a

P

J 0,j

__

rji

V .

1 a

)

for our choice of orientation lp,p it is also true that v 0/

a v/9, j

v _ , v ta

•

Inserting this in (5.27), we obtain ( t f a l'V • O

= Z j V j J j ■t j ) = ( x j ■taP) Z i Vi­

lli view of 5.2-f, X ; Vj = 1, so that ( S j'y J

■t j ‘) = ( x j • t j ) .

On the other hand, {SaJ D j x J

• t al") =

( D j x j • * • • • > Tpn~' >

• > 7V),

and that S J z J does not vanish on Tpp. Therefore, S j ' i y = T J G Ka \

K0a

•

§5]

A

C O M BINATORIAL

65

LEMMA

The simplex S al0lT?1q lias the form S j ' T r f = ( s j ri y > ■■■ > S j T f ) , where S a0Tpp = Tap. Let T an be the carrier of 7 / in K a , and set 7V'° = S j T 0n Since S j is a normal mapping, T„" > 7Y'°, so that 0(1 r an° I; 7 ’/ ) c 0(1 r a” I; T „ r ).

Ihe vertices of S a 0lT p 9 are the centroids of the simplexes 7’ “0 _ 1 a —

V 0 r n n v. I P d_

v.

‘

ti P rn V J p =

rli V / « ,

and are therefore contained in the convex set 0 ( [ 7 V ° |; r / ) C 0(1 Tan I; 7’/ ) ; hence the closed simplex Sai^Tpi9is also contained in the set 0( | T„n \; Tttp) : (5.311)

SaiSiT0l“ C 0(1 r y I; Tap)

(where S ai0iTpi9 denotes the closure of S ai0lTpi9). It remains to be proved that (5.312)

f y C Z 0 ( \ T an \ ; T ap).

Letting Tan' (1 < i < q) stand for the carrier of the simplex Tpn~’ in K a (and recalling that Tan is the carrier of Tpn), we see that rn

(5.313)

'or

n

n.

rn

-* rt

•» l

>

••• >

Ty.

Since S j is a normal mapping, T ap = S a0Tpp is a face of 7Y'1’; hence, by (5.313) , (5.314)

ry > ryi> ■■> ry > ry

It follows that all the simplexes of (5.314) are contained in 0 (| Tan ]; Tap) ; in particular, 0 (| 7V |; T av) contains the centroids of the simplexes

i.e., the vertices of Tp,q. Since 0 (| Tan |; Tap) is a convex set, and contains all the vertices of Tp9, it also contains the closed simplex Tp9. This proves 5.31. We shall now complete the proof of Lemma 5.1. In view of 5.31, all the skeletons of the prisms spanned by | Tp9 | and S a 0]\ Tp9 \ are contained in the convex set 0(1

ry |; rap)c

| | / v a | | \ ! | / u „ I!.

Since 0 (| Tan |; Tap) is convex, it also contains the closed convex hulls of

GC)

V-G U O U PS

OF COMPACTA.

A LEX A X D ER-PO N TU Y A G IX

D U A LITY

[CH.

X IV

all the skeletons of the prisms mentioned above. The union of the closed convex hulls of the skeletons of the prisms spanned by all the simplexes Tpj9 satisfying the conditions of 5.31 is a compactum (even a polyhedron) TT contained m G a = || K a || \ || K 0a ||, and n contains the prism spanned by the cycles zpx9 and tiai lzpi. Therefore, q

c i & l q

2/31 ~ Oa| Z.3i

t t

Oil IT

(in the sense of XI, 4.0), i.e., (5.32)

sefDfSfz* ~

on IT,

because of the definition of z^' in 5.31. By (5.2), we have S l a l V D f S p V = SaiaDa9ZaP. Furthermore, since saiaDa9zap is contained in n, r\ q tx S01a lT\) aqZaV = S/Ha r l S„1ct Da Zar> ~ $ala D a1SpfDp9Spazap can be replaced by sp]aDa9zaP in (5.32). We then obtain S^Df Sf Za* - S^aDa9ZaP - 0 Oil II. As we saw in 4.2, there is a continuous mapping C of Gp onto j, Qpi || which leaves all the points of |j Qpi |! fixed, and which consequently maps the cycle spi Dp9Spazap — spiaD a9zaP on Qpi into itself. Therefore, (5.33)

spfDp9Spazap - sp°Da9ZaP ~ 0

in C(TT) c |j Q# Jj.

But Spi0Dp9Spaz ap -

splaD a9z ap € 7 ^ 9{Qin),

so that (5.33) implies (by XI, Corollary 4.01) that s e f D p ' S p V - s p f DJ z S ~ 0

in Qn .

This completes the proof. §6. The spectrum 4C(i of the set 4> §6.1. The coverings and u>a . Retaining the notation of 4.1, we denote by Qa the open covering of S n consisting of the open stars (IV, 3.2) of K a : n . = IOi“. ■■■ We know that K a is the nerve of the covering . Suppose that 0 i“, • • • , ()s0(I“ are those elements of V.a which intersect 4>. We set (0.11)

( l < f < s 0a).

The open sets a", • • • , oSOu" of form a covering of which we denote by co„ .

§0]

TH E

SPECTRUM

JC0 OU

A

07

CO .M PA CTU M

0.12. The complex /v0„ is the nerve of the covering u„ . For, the vertices of 7\0a are precisely those vertices of K a "whose open stars (relative to I \ a) intersect $. In other words, the vertices of K 0a cor­ respond (1— 1) to the elements of ua (a vertex e of 7v0a corresponds to the element $ n 0 { K a ; c) of ua [where 0 { K a ; c) is the star of c in 7va], [The elements of ua form an indexed set (I, 1.3); some of them may coincide geometrically.] Moreover, the intersection of the open stars of a set of vertices e,f, • • • , cira of K a is nonempty if, and only if, there is a simplex Ta with vertices c,-f, • • • , e , f in K a ; and then 0 ( K a ; a f ) n • • • n 0 ( K a ; c,ra) = 0 { K a ; T a). Therefore, the nerve of coa consists of precisely those simplexes of K a whose stars intersect 4>, i.e., those which are faces of simplexes of K a whose inter­ section with is nonempty. But then K 0a is the nerve of ua . It is clear that if (8 > a, then 9.p is a refinement of 9 a and consequently up is a refinement of u a . On the other hand, Condition 2° of 4.1 implies that there is a simplex Too, t Kaa such that the diameter of T n 7\ia is greater than twice the diameter of an arbitrary Tp 6 Kp . Hence, there is an element of ua which is not contained in any element of 9.p and therefore in no one single element of cup . Hence coa is not a re­ finement of Up . We have proved (i.13. If /3 > a, up > ua . §6.2. The spectrum Jt0 of the compactum T. Let us assign to each element O f H of the covering ha+1 a definite element Of = S aa+I (Oja+i) of 9 a which contains O f ' 1. This assignment defines a projection of (2 a+i into 9.a and also a projection ,S'oan,“+1 of w«h into ua : the element o f = n 0; “"H of oi oja4-i wafl is w assigned 0, a H_ £*+1 = (I>n S aa H0 f of ua ; since O f 3 O f 1, the element o f = S0a.....o j a —\ , a-Fl Oi 3 Oj For p > a + 1, we set op ii .(3—1 dfi 0. a+1 ■■■ $ o ^“1ryp— c< i ,. o &— _ Oa y “J $ I)a $0,0-1 0 0 -2 $ Oor o^.-2 »'a .

We have therefore defined for every pair of natural numbers a and p, 0/3 into u a , such that 0 > a, projections S j of P-p into 9.a and S0a°P of C up 0 7 s' 00^a701007 •y S ap S p Oa 7ej ai (0-21) for 7 > p > a. The projections of the coverings induce simplicial mappings (denoted by the same letters) of the nerves of the coverings, i.e., a projection S j of K^ into K a and a projection S f of 7C,?info 7w . with and S f coinciding on I\ o0 .

G8

V-g roups o r

compacta . alex axd er - poxtryagix duality

[ch .

xiv

The identity (0.21) shows that these projections satisfy Condition 2° of 1.1; Condition 1° of 1.1 is also satisfied, since the simplicial mappings S f , Soa 0 are induced by the corresponding projections of the coverings Pp , cop into P.a , • Hence, 'the sequence of complexes [Koa]

(« = 1,2, ■•■)

together with the projections Soa13just now defined on them form a spectrum of a, then Sj ep is a vertex of the carrier of e.p in K a . For, according to the definition of 0( Kp ; ep) C 0 ( K a ; Sjep), e.p e II T a II C 0 ( K a ; sj ep) , so that Sjep is a vertex of the carrier T a 6 K a of ep . C orollary 0.32. For arbitrary Tp £ Kp , the simplex S a0T3 is a face of the carrier of Tp in K a\ In fact, if Tp C T„ £ K a , then the carrier of each vertex of Tp is a face of T a ; hence, by 0.31, the image of each vertex of Tp is a vertex of T a , which proves 0.32. §7. Further auxiliary propositions §7.1. Commutativity of the operators Sp" and VE (see VII, 5.0 for the definition of E). Let us consider the complexes K a , Kp and their subeoinplexcs Ki)a and K 0p . The projection S j of Kp into K a induces a homomor­ phism Spa of Lp( I \ a) into L ’f Kp) , and the same projection restricted to the complex K 0a Cl I ( a induces a homomorphism Sop'a of L p( K 0a) into L p( K 0p). Then 7.1. The. relation (7.1)

SpaVEz0av -

TES»p0az»aP in K a \

K 0a

holds for every z0aP £ Zvp( KDa). Proof. Since the operators Spa and V commute [see IX, (0.2)], we may rewrite (7.1) as VSpaE.z0„v — ? E S w az»„p in K a \

K 0a ,

or in the form (7.11)

V(SpaEz0ap - E S ^ z J f i - 0

in Ka \

K 0a •

A U X ILIA RY

09

PR O PO SITIO N S

But (7.11) obviously follows from (7.12)

Si,aEz0ap - E S ^ az0ap G Lv{ K a \

K 0a).

Wc shall pro\'e (7.12). For arbitrary 7’/ £ K ti and arbitrary orientation / / of 7 / , ( Sf Ez o S • t f ) = (Ez0ap • S a% p)-, hence if S / T p p G Ko