Combinatorial Topology Volume 2 [02] 0486401790, 9780486401799

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P.S. Aleksandrov

COMBINATORIAL TOPOLOGY Volume 2

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COM BIN A TORI A L TOPOLOGY VOLUME 2 THE BETTI GROUPS

OTHER GRAYLOCIC PUBLICATIONS

KHINCHIN: Three Pearls of Number Theory PONTRYAGIN: Foundations of Combinatorial Topology NOVOZHILOV: Foundations of the Nonlinear Theory of Elasticity ALEKSANDROV: Combinatorial Topology, Vol. 1 PETROVSKlI: Lectures on the Theory of Integral Equations KOLMOGOROV and FOMIN: Elements of the Theory of Functions and Functional Analysis, Vol. 1. Metric and Normed Spaces

COMBINATORIAL TOPOLOGY VOLUME 2 THE BETTI GROUPS

BY

P. S. ALEKSANDROV

G R A Y L O C K

P R E S S

ROCHESTER, N. Y. 1957

'J’K A N S J ^ A T K I > J ’lfOJVI

'1’J I 10 F I R S T

R U S S IA N

K D IT IO N

BY HORACE

KOMM

C o p y rig h t, 1957, by GRAYLOCK PR ESS R o c h e s te r , N . Y .

All rights reserved. T his book, or p arts thereof, may not be reproduced in any form, or tra n s la te d , w ith o u t p erm is­ sion

in

w ritin g

from

th e

p u b lish ers.

L i b r a r y o f C o n g r e s s C a t a l o g C a r d N u m b e r 5 6 —1 3 9 3 0

Manufactured in the United States of America

CONTENTS PART TH REE TH E B ETTI GROUPS Chapter

VII. C haixs . T iie Opehator A 1. O rientation.................................................................................... 1.1. Orientation of the space R n................................................ 1.2. Orientation of a simplex and of a skeleton...................... 1.3. The body of an oriented simplex...................................... 1.4. Extension of an orientation tn to an orientation R n. The product orientations tnR n and tinUn.......................... 1.5. The orientation (e0C-1) ........................................................ 1.6. Affine images of orientations.............................................. 2. Intersection Number of Planes and Simplexes........................ 2.1. Intersection number of planes............................................ 2.2. Intersection number of simplexes...................................... 2.3. Intersections and simplicial mappings.............................. 3. Incidence Num bers....................................................................... 3.1. Definition of the incidence num bers................................. 3.2. Properties of the incidence num bers................................. 4. Cell Complexes; o-complexes...................................................... 4.1. Definition of a-complexes and cell complexes.................. 4.2. The incidence matrices of a cell complex......................... 5. C hains............................................................................................. 5.1. Definition of a chain............................................................ 5.2. Some remarks on chains...................................................... 5.3. Monomial chains. Chains as linear form s........................ 5.4. Chains of a simplicial complex........................................... 5.5. The scalar product of chains.............................................. 5.6. Extension of chains; restriction of chains to a subcom­ plex. The operators and ........................................... 6. The Lower Boundary Operator (The Operator A)................. 6.1. Definition of the A-boundary............................................. 6.2. Examples of chains and their boundaries......................... 6.3. Cycles; chains homologous to zero; the groups Z T($) and IITm ............................................................................... 6.4. Homologies. The symbol Linear independence of chains with respect to homology....................................... V

2 2 3 5 5 0 7 8 8 10 11 11 11 12 13 14 16 18 18 20 2L 22 23 23 24 24 26 29 30

CONTENTS

VI Chapter

0.5. 6.6. 7. The 7.1. 7.2. 7.3.

Restricted chains and cycles.............................................. Extension of chains and cycles......................................... Fundamental Case: $ is an a-complex............................ The fundamental formula AA.Tr = 0 ................................ Closed and open subcomplexes of an a-complex............. Weak homology of integral cycles; the dual coefficient dom ain.................................................................................. 8. Simplicial Images of Chains....................................................... 8.1. Simplicial images of oriented simplexes............................ 8.2. The homomorphism S f of the group L (Kp) into the group I f (K a) induced by a simplicial mapping S a of a complex Kp into a complex K a.......................................... 8.3. Commutativity of the operators A and S a .................... 8.4. The case of open subcomplexes......................................... 9. Auxiliary Constructions.............................................................. 9.1. Cone over a chain................................................................ 9.2. Application of the constructions of 9.1............................ 9.3. Prism over a chain.............................................................. 9.4. Application to simplicial mappings................................... Addendum. The a-complex of the Oriented Elements of a Polyhedral Complex.................................................................... VIII. A-groups

of

31 32 34 34 35 35 36 36

37 38 39 40 40 41 43 45 47

C omplexes (L ower B etti or H omology G roups)

1. Definitions. Examples. Simplest General Properties.............. 1.1. Definition of the group Ar($, 91)...................................... 1.2. The groups An( $ n, 91)......................................................... 1.3. The groups A0(/C, 91)........................................................... 1.4. Simplest examples of the groups Ar .................................. 1.5. Some elementary u-complexes and their Betti groups. . 1.6. The group A00(/v, 91)........................................................... 1.7. Decomposition of the group Ar($, 9f) into a direct sum over the components of the complex £ ............................ 1.8. The homomorphism of the group AT(Kp, 91) into A\ K a , 9f) induced by a simplicial mapping S j of a sim­ plicial complex Kp into a simplicial complex K ................ 2. The Groups Anr(^ )...................................................................... 2.1. The torsion groups.............................................................. 2.2. The groups A00rGG.............................................................. 2.3. Finite a-complexes. Homology bases................................ 2.4. The Euler-1’oincare formula for a finite n-dimcnsional a-complex..............................................................

50 50 50 50 53 (il 65 66

67 OS 08 09 70

CONTENTS

Vii

3. Pseudomanifolds........................................................................... 3.1. Pseudomanifolds.................................................................. 3.2. Orientablo pseudomanifolds............................................... 3.3. The groups Amn( K n) of a nonorientable n-dimensional pseudomanifold. Disorienting sequences.......................... 4. Addenda and Exam ples.............................................................. 4.1. The Betti groups of the complexes | T n | and T n = \ T n \ \ r n............................................................................ 4.2. Surfaces.................................................................................. 4.3. Simple pseudomanifolds. Elementary triangulations. . 4.4. Applications to projective spaces...................................... 5. Simplicial mappings of pseudomanifolds................................. 5.1. The degree of a m apping................................................... 5.2. The original definition of the degree of a simplicial m apping.................................................................................

72 72 74

Chapter

77 79 79 80 81 83 86 86 86

IX . T h e O p e r a to r V and t h e G roups Vr (.$, 31). C a n o n ic a l B a ses. C a lc u la t io n o f t h e G roups Ar ($, 31) and Vr (S?, 3f) B y M ea n s o f t h e C roups A0r (Si)

1. The 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

Operator V............................................................................. Definition of the chain VxT................................................. The chain V.rr as a linear fo rm .......................................... Dualit3r of the operators A and V...................................... The groups Z vT(Si, 31), / / / ( $ , 31), Vr(tf,31)...................... Chains restricted to a subcom plex.................................... The groups V°($, 31) = Z v°(Si, 31)................................... The groups Vn( K n, J) of n-dimensional pseudomani­ folds ........................................................................................ 2. Bases of the Modules LS(Si)....................................................... 2.1. Preliminary rem arks............................................................. 2.2. Dual bases of L 0T(Si)............................................................. 2.3. The elements of the group IS (Si, 31) expressed in terms of a basis of the module LS(Si).......................................... 3. Canonical Systems of Bases. The Groups V0’’(^').................. 3.1. Preliminary rem arks............................................................. 3.2. Canonical bases of the groups Z S ..................................... 3.3. Canonical homology bases................................................... 3.4. A system of canonical bases of the groups I S .................... 3.5. A system of V-bases for $I; the groupsV0r( ^ ) .................... 4. Calculation of the Groups Ar($, 31) and Vr($, 31) By Means of the Groups A0r($, 31)................................................................

90 90 92 92 93 94 94 95 97 97 98 99 100 100 100 101 102 105 109

Vlll

CONTENTS

Chapter

4.1. Calculation of the groups Ar($, §1)................................... 4.2. Calculation of the groups Vr($, 51).................................. 4.3. The coefficient domains J, 9i, 91i....................................... 4.4. The groups Amr($) and Vmr( $ ) ......................................... 4.5. Integral chains and homologies (mod m )......................... 5. Calculation of the Groups Ar($, 51) and Vr(l?, 51) By Means of the Groups Ar($, §Ri) and Amr( £ ) .......................................... 5.1 ................................................................................................. 5.2 ................................................................................................. 6. The Homomorphism Spa of L T( K a , 51) Into L \ K &, 51) In­ duced by a Simplicial Mapping S ap of a Complex Kp Into a Complex K a .................................................................................. 6.1. Definition of the homomorphism Spa................................ 6.2. The commutativity of the operators V and Spa............. 6.3 ................................................................................................. X. I nvariance

of the

109 Ill 112 113 114 117 117 118

119 119 120 120

B etti G roups

1. Formulation of the Invariance Theorems................................ 125 1.1. Definition of the numbers F(4>)......................................... 125 1.2. Definition of the groups ©r(4>)............................................ 126 1.3. Formulation of the invariance theorem for the Betti numbers and groups............................................................. 126 2. Subdivisions of Chains. Fundamental Systems of Subcom­ plexes and Chains. Invariance of the A- and V-groups un­ der Elementary and Baryccntric Subdivisions...................... 127 2.1. The isomorphism ............................................................ 127 2.2. Fundamental systems of subcomplexesof a complex K . . 129 2.3. Fundamental systems of chains......................................... 131 2.4. The a-complex defined by a given fundamental system of chains................................................................................ 135 2.5. The isomorphism j3 of V ($) into //(fi^ ).......................... 137 2.6. The invariance of the Betti groups under elementary and baryccntric subdivisions of K ..................................... 139 3. Normal and Canonical Displacements in Polyhedra.............. 139 3.1. Normal displacements of subdivisions of triangula­ tions ....................................................................................... 139 3.2. Examples of normal homomorphismsS j and Spa. . . . 142 4. Canonical Systems of Bases for Subdivisions K$ of a tri­ angulation K a . The Homomorphism Spa Dual to a Nor­ mal Homomorphism S j ............................................................ 144 4.1. A canonical system of bases for .................................. 144

CONTEXTS

ix

Chapter

4.2. Normal homomorphisms in canonical bases.................. 4.3. The homomorphism dual to a normal homomorphism. . 5. Complexes K(R, «). Small Displacements in Polyliedra and Compacta. The Pflastersatz and the Invariance of the Betti Num bers.......................... 5.1. The complex K(R, e); e-chains of ametric space R. . . . 5.2. e-displacements..................................................................... 5.3. Canonical displacements..................................................... 5.4. The numbers tj(7v). Canonical displacements in polyhedra...................................................................................... 5.5. The Pflastersatz. Invariance ofthe Betti num bers........... 6. Invariance of the Betti Groups.................................................. 6.1 ................................................................................................. 6.2. Invariance of the Betti groups for polyhedral complexes 7. Invariance of Pseudomanifolds.................................................. 7.1. Formulation of the theorems.............................................. 7.2. Proof of Theorem 7.14.........................................................

146 147

148 148 149 149 150 151 153 153 154 155 155 156

X I. T he A-groups of C ompacta

1. Definition of the groups Ar(4>, 21)............................................... 1.1. Proper cycles......................................................................... 2. Lemmas on e-displacements and e-homologies...................... 2.1. Prisms and e-displacements................................................ 2.2. The case of a polyhedron 4> = || K a ||.......................... 3. The Homomorphism of the Groups Ar(4>) Induced by a Continuous Mapping of a Com pactum ..................................... 3.1. The continuous image of a proper cycle.......................... 3.2 .................................................................................................. 3.3. Homology classification of mappings................................ 3.4. Deformation of a continuous mapping of a proper cycle. Deformation of a proper cycle........................................... 4. The Fundamental Theorem on the Ar-groups of Polyhedra. 4.1. Fundamental Theorem 4 .1 ................................................. 4.2. Construction of the homomorphism S a* of A$>r into Aar ............................................................................................ 4.3. The mapping S a$ is a mapping onto Aar........................... 4.4. The homomorphism Sa* of A$r onto A j is an isomor­ phism ....................................................................................... 4.5. Rules for finding the images of the isomorphisms S a* and ............................................................................ 4.6. Cycles z j 6Z j and homologies in= || K a | | ...................

158 158 159 159 161 163 163 164 164 166 166 166 166 167 168 168 169

CONTENTS

X

Chapter

4.7. The image of a cycle z j 6 Z J under a continuous map­ ping C of a polyhedron = || K a || into a compactum ...................................................

178 178 178 180 181 182 182 184

185 186 186 187 187

CONTENTS

xj

Chapter

3.2. (46 'I'')-homologous and (46 46)-homotopic mappings; (46 46)-defonnations............................................................ 3.3. Deformation of a relative cycle of ................................. 4. The Groups A,6(r) of Polyhedra and 4 '............................. 4.1. Introductory rem arks.......................................................... 4.2. The fundamental theorem .................................................. 4.3. The homomorphism Ca' “ of Ar0r = A ( K a \ K y a , W) into Ar'a'T = Ar(/v«« \ AV«' , ?t) induced by a (T, T')mapping CV*........................................................................ 4.4. Definition of the homology dimension of a polyhedron. Another proof of the invariance of the dimension num­ b e r.......................................................................................... 4.5. The definition of the homology dimension of a compac­ tio n ........................................................................................ 5. Pseudomanifolds With Boundary.............................................. 5.1. Orientation of a pseudomanifold with boundary.......... 5.2. Introductory remarks; definition of the degree of a continuous mapping of a pseudomanifold with bound­ a ry .......................................................................................... 5.3. Some properties of the degree of a m apping.................. 5.4. Examples.............................................................................. 0. The Groups A/(4>) (The Local Ar-groups of a Compactum 4>). 6.1. Definition of the groups Apr().......................................... 6.2. The local character of the groups A / .............................. 7. The Local A-groups of Polyhedra............................................. 7.1. Notation and introductory rem arks................................. 7.2. The fundamental theorem .................................................. 7.3. Application to the invariance of pseudomanifolds.......... A ppendix 2 .................................................................................................. L ist of S ym bols .......................................................................................... I n d e x .............................................................................................................

188 I89 19o 190 19q

192

192 193 193 193

194 195 195 198 198 199 20l 201 203 209 210 238 241

Part Three

THE B ETTI GROUPS The Betti groups (defined in Chapter V III) are the central concept of combinatorial topology. All of P art Three, as well as much of the sequel, is devoted to their study. The underlying algebraic theory of combinatorial topology is developed in Chapter VII. It is used, in particular, to define and investigate the Betti groups themselves. The algebraic apparatus is made up of the two concepts: a chain and a boundary operator A. These two ideas are fundamental to Chapter VII. These in turn depend on the notions of orientation (discussed at the beginning of Chapter ATI) and of an a-complex. The latter is a natural generalization of the set of all oriented simplexes of a triangulation. Having introduced all the auxiliary algebraic concepts in Chapter VII, we deal in Chapter ATII with the definition and elementary theory of the “ lower” Betti groups (homology groups) or, as we shall call them here, the A-groups of triangulations (and, in general, of a-complexes). Chapter ATII concludes with an investigation of orientable and nonorientable pseudo­ manifolds which, in addition to other examples, serve to illustrate the theory. Chapter IX deals with more complicated problems of the theory of Betti groups. First, the “ upper” Betti groups or V-groups (cohomology groups) are introduced. Then the Betti groups are studied by means of canonical bases which, in particular, enable us to derive a relation among the Betti groups over various coefficient domains. In Chapter X we prove the invariance of the Betti groups, th at is, th at all the (topological) triangulations of a polyhedron (or of homeomorphic polyhedra) are isomorphic. In Chapters XI and X II the concept of A-group is extended from poly­ hedra to arbitrary compacta. It should be noted, however, th at this gen­ eralization can be effected in a completely adequate fashion only by means of the theory of topological groups, which is beyond the scope of this book. I have succeeded in avoiding this difficulty, but only by defining the V-groups, and not the A-groups, of arbitrary compacta. This is done, but considerably later, in Chapter XIAC Chapter X II, among other things, contains an account of the local A-groups. These are used in X III, 1.1, to give a simple invariant definition of /i-manifolds. 1

Chapter V I I

CHAINS. THE OPERATOR A §1. Orientation §1.1. Orientation of the space R n. The concept of an oriented or directed segment will already be familiar to the reader who has had a course in elementary algebra. In this section the notion of an oriented segment will be generalized to n dimensions. We shall call a collection of n + 1 linearly independent points of R n written in a definite order an ordered skeleton of R n. According to this defi­ nition two different ordered skeletons may consist of the same points, differing from each other only in the order of these points. We recall (Ap­ pendix 1, 1.5) that there is precisely one affine mapping of R n onto itself which carries a given ordered skeleton ] e0 , ex , • • • , en | into a preassigned ordered skeleton | e'0 , e\ , • • • , e'n |, that is, which maps the points