Seminar on Transformation Groups. (AM-46), Volume 46 9781400882670

Table of contents :
TABLE OP CONTENTS
INTRODUCTION
CHAPTER I: COHOMOLOGY MANIFOLDS
§1. Preliminaries
§2. Local Betti Numbers Around a Point
§3. The Notion of Cohomology Manifold
§4. Some Properties of Cohomology Manifolds
§5. Appendix on Cohomological Dimension
CHAPTER II: HOMOLOGY AND DUALITY IN GENERALIZED MANIFOLDS
§1. Homology Groups for Locally Compact Spaces
§2. Duality in Generalized Manifolds
§3. Existence of Fundamental Sheaves for Homology
§4. Closed Subsets of Euclidean Spaces
§5. Complements
CHAPTER III: PERIODIC MAPS VIA SMITH THEORY
§1. The Leray Spectral Sequence
§2. Transfer Homomorphism
§3. An Euler Characteristic Formula
§4. The Smith Sequences
§5. Orbit Spaces of Finite Groups b
CHAPTER IV: THE ACTION OF 2L OR T1 : GLOBAL THEOREMS
§1. Transformation Groups BG
§2. Some Remarks on the Cohomology of XG
§3. The Space
§4. The Fixed Point Set of a Prime Period Map in a Cohomology Sphere
§5. The Action of the Circle Group
§6. The Quotient of a Cohomology Sphere
CHAPTER V: THE ACTION OF OR T1 : LOCAL THEOREMS
§1. Conservation of Cohomological Local Connectedness
§2. The Fixed Point Set of a Prime Period Map in a Cohomology Manifold
§3. The Fixed Point Set of a Toral Group in a Cohomology Manifold
§4. Remarks on Local Groups of the Quotient Space
CHAPTER VI: ISOTROPY SUBGROUPS OP TORAL GROUPS
§1. Introduction
§2. A Regular Convergence Theorem
§3. Two Lemmas
§4. Proof of Theorem 1.2
CHAPTER VII: FINITENESS OF NUMBER OF ORBIT TYPES
§1. Preliminary Remarks
§2. Statements of the Main Results
§3. Proof of the Main Theorem
CHAPTER VIII: SLICES AND EQUIVARIANT IMBEDDINGS
§1. Notation and Preliminaries
§2. Orbit Types
§3. Slices
§4. Equivariant Imbeddings in Euclidean Space
CHAPTER IX: ORBITS OF HIGHEST DIMENSION, by D. Montgomery
§1. Introduction
§2. The Set B
§3. The Set D
§4. The Set B H D
§5. Conditions Under Which dim2D

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Annals of Mathematics Studies Number 46

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by

H erm an n W eyl

3. Consistency of the Continuum Hypothesis, by 6. The Calculi of Lambda-Conversion, by 10. Topics in Topology, by

K u rt G odel

A lon zo C hurch

S o lo m o n L e fsch e tz

11.

Introduction to Nonlinear Mechanics, by N.

15.

Topological Methods in the Theory of Functions of a Complex Variable, by

16. Transcendental Numbers, by 17.

K ryloff

and N.

S. B ochner

and K.

L ia p o u n o f f

C h an d r ase kh ar an

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. 21. Functional Operators, Vol. I, by

J ohn vo n N e u m a n n

24.

Contributions to the Theory of Games, Vol. I, edited by

25.

Contributions to Fourier Analysis, edited by A. d e r o n , and S . B o c h n e r

26. A Theory of Cross-Spaces, by

L e fsc h e tz

J oh n v o n N e u m a n n

22. Functional Operators, Vol. II, by

27.

H . W . K uhn

Zy g m u n d ,

W.

and

A . W . T ucker

T r a n su e , M . M o r se ,

28. Contributions to the Theory of Games, Vol. II, edited by

31. 32.

Polya

and G.

H . W . K uhn

Contributions to the Theory of Riemann Surfaces, edited by L . Order-Preserving Maps and Integration Processes, by E d w a r d

Cal­

S zego

and

Curvature and Betti Numbers, by

K . Y ano

and S.

A h lfo rs

A . W . T ucker

34. Automata Studies, edited by C. E.

S h an n on

and

L e fsch e tz

et al.

J. M c S hane

B ochner

33. Contributions to the Theory of Partial Differential Equations, edited by and F. J o h n 35. Surface Area, by

A. P.

R obert S ch atten

Isoperimetric Inequalities in Mathematical Physics, by G.

29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. 30.

M a r st o n M o r se

C a r l L u d w ig S ie g e l

Probleme General de la Stabilite du Mouvement, by M. A.

19. Fourier Transforms, by

B o g o l iu b o f f

L . B ers,

S.

Bochner,

J. M c C arth y

L am b e r t o C e sar i

36.

Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S.

37.

Lectures on the Theory of Games, by

H arold

W.

K uhn.

38. Linear Inequalities and Related Systems, edited by

L e f sc h e t z

In press

H . W . K uhn

and

A . W . T ucker

39. Contributions to the Theory of Games, Vol. Ill, edited by P. W o l f e

M . D r e sh e r , A . W . T u c k e r

40. Contributions to the Theory of Games, Vol. IV, edited by R.

D un can L uce

41.

Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S.

42.

Lectures on Fourier Integrals, by S.

A . W . T ucker

L e fsch e tz

B ochner

43. Ramification Theoretic Methods in Algebraic Geometry, by S. A b h y a n k a r 44. Stationary Processes and Prediction Theory, by H. F u r s t e n b e r g 45. Contributions to the Theory of Nonlinear Oscillations, Vol. V, edited by S. 46. Seminar on Transformation Groups, by A. 47. Theory of Formal Systems, by R.

and

Borel

Sm ullyan

et al.

and

L e fsc h e tz

SEMINAR ON TRANSFORMATION GROUPS BY

Armand Borel W ITH CONTRIBUTIONS BY

G. Bredon E. E. Floyd

D. Montgomery R. Palais

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 19 6 0

Copyright © 1 9 60 , by Princeton University Press

All Rights Reserved Lo C. Card 6 0 - 1 2 2 2 5

Printed in the United States of America

TABLE OP CONTENTS Page 1

INTRODUCT ION CHAPTER Is

CHAPTER II:

COHOMOLOGY MANIFOLDS, by A. Borel §1 . Preliminaries §2 o Local Betti Numbers Around a Point §3« The Notion of Cohomology Manifold §*u Some Properties of Cohomology Manifolds §5* Appendix on Cohomological Dimension HOMOLOGY AND DUALITY IN GENERALIZED MANIFOLDS by A. Borel §1 o Homology Groups for Locally Compact Spaces §2 o Duality in Generalized Manifolds §3» Existence of Fundamental Sheaves for Homology §4. Closed Subsets of Euclidean Spaces §5- Complements

CHAPTER III: PERIODIC MAPS VIA SMITH THEORY, by E. E. Floyd §1 o The Leray Spectral Sequence §2 . Transfer Homomorphism §3 « An Euler Characteristic Formula §4. The Smith Sequences §5* Orbit Spaces of Finite Groups CHAPTER IV:

CHAPTER V:

THE ACTION OF 2L OR T 1 : GLOBAL THEOREMS, by Ac Borel p §1 o Transformation Groups §2 . Some Remarks on the Cohomology of §3 * The Space §4. The Fixed Point Set of a Prime Period Map in a Cohomology Sphere §5 . The Action of the Circle Group §6 . The Quotient of a Cohomology Sphere by Zp or T 1 THE ACTION OF OR T 1 : LOCAL THEOREMS, by A. Borel p §1 . Conservation of Cohomological Local Connectedness §2 . The Fixed Point Set of a Prime Period Map in a Cohomology Manifold §3* The Fixed Point Set of a Toral Group in a Cohomology Manifold §4. Remarks on Local Groups of the Quotient Space

5 5 7 9 10 18

23 23 25

27 30

31 35 35 37 39 40 b3 ^9 b-9 50 52

55 6o 63 67 67

lb 80 83

CONTENTS Page CHAPTER VI:

CHAPTER VII:

ISOTROPY SUBGROUPS OP TORAL GROUPS, by E. E. Ployd §1 . Introduction §2 . A Regular Convergence Theorem §3» Two Lemmas §bo Proof of Theorem 1 . 2 FINITENESS OP NUMBER OF ORBIT TYPES, by G. E. Bredon §1. Preliminary Remarks §2. Statements of the Main Results §3« Proof of the Main Theorem

CHAPTER VIII: SLICES AND EQUIVARIANT IMBEDDINGS, by R. So Palais §1. Notation and Preliminaries §2 . Orbit Types §3 * Slices §4. Equivariant Imbeddings in Euclidean Space CHAPTER IX:

CHAPTER X:

ORBITS OF HIGHEST DIMENSION, by D. Montgomery §1 o Introduction §2. The Set B §3 o The Set D §4o The Set B H D §5- Conditions Under Which diti^D< n - 2 §6. Remarks on the Differentiable Case THE SPECTRAL SEQUENCE OF ABIFILTERED MODULE, by A. Borel §1. Spectral Sequences §2. The Notion of Bifiltration §3* The Terms EQ and E 1 §4. Two Further Assumptions §5« The Term E2 §6. Homomorphisms §7* Remarks

CHAPTER XI: THE SPECTRAL SEQUENCE OF FARY, by A.Borel §1. Sheaves §2* Continuous Maps §3* The Spectral Sequence of Fary §4. Locally Compact Spaces

85 85 86 89 90 93 93

93 95 101 101

10k 105

112 117 117 117 121 123 125 128

133 133 13^ 136 139 1^1 1*4-2 1^ U5 1^5 1^6 1^9 152

CONTENTS Page CHAPTER XII:

FIXED POINT THEOREMS FOR ELEMENTARY COMMUTATIVE GROUPS I, by A. Borel §1. Some Fiberings §2- The Topology of the Leray Sheaf §3» Fixed Point Theorems §4 0 Applications to Projective Spaces §5 ® Applications to Compact Lie Groups §6. Applications to Homologically Kahlerian Manifolds

CHAPTER XIII: FIXED POINT THEOREMS FOR ELEMENTARY COMMUTATIVE GROUPS II, by A. Borel §1o Notation §2. Cohomology Spheres §3° Some Local Concepts §4o Cohomology Manifolds CHAPTER XIV:

ONE OR W O CLASSES OF ORBITS, by A. Borel §1« One Class of Orbits on Spheres §2. A Sufficient Condition for the Existence of Fljced Points on Rn §3« Two Classes of Orbits on Euclidean Space

16o

162 167 169 1 70

173 173 175 180

181 185 185 188

190

CHAPTER XV:

on

FIXED POINT SETS AND ORBITS OF COMPLEMENTARY DIMENSION, "by Go E. Bredon §1o Introduction §2 0 Preliminary Results A Theorem on Cross-Sections §4 0 Case I. rank (G) = rank (H) §5o Case II0 rank (H) < rank (G) §6. Rank One Case §7. A Local Cross-Section for the Orbits of G Near F Conclusion of the Proof of Theorem 1*4 CHAPTER XVI: REMARKS ON THE SPECTRAL SEQUENCE OF A MAP, by A. Borel §1* Cohomology with Compact Supports §2 o The General Case §3* Inductive Limits §4* On the Leray Sheaf §5« A Case of Validity-for the Kunneth Rule

157 157

195 195 1 98

207 21 1

214 215

CO

2 23 230 233 233 235

239 241 243

SEMINAR ON TRANSFORMATION GROUPS

INTRODUCTION In this book, a transformation group is a compact Lie group G acting on a topological, usually locally compact, space X. Transforma­ tion groups are discussed mainly from the point of view of algebraic topology, and the problems most often center around relations between topological properties of G, X, the fixed point set, the orbits, and the space of orbits X/G. A typical topic along these lines, and chrono­ logically one of the first ones, is P. A. Smith’s theory of prime period maps of homology spheres. Apart from some modifications and additions, this book consists of the Notes of a Seminar held at the Institute for Advanced Study in 1 9 5 8 -5 9 . Familiarity with the basic notions in the theory of transforma­ tion groups and with algebraic topology is assumed. As to the latter, the results concerning sheaves, spectral sequences and fibre bundles, for which references to the expositions of Cartan, Godement, Steenrod or the under­ signed can be made, are usually taken for granted. However, certain topics for which this is not possible are given an independent discussion. As a result, five chapters (I, II, X, XI, XVI) are concerned with questions of algebraic topology and make no mention of transformation groups. We first briefly summarize those. Chapter I is devoted to cohomology manifolds. These are spaces which have the local cohomological properties of manifolds. The definition adopted here is equivalent to that of a locally orientable generalized manifold in the sense of Wilder. Chapter II introduces a homology theory for locally compact spaces and uses it in order to derive a Poincare duality theorem for cohomology manifolds. Chapter X is purely algebraic, and is concerned with the spectral sequences attached to a differential module endowed with two filtrations. The discussion is carried only as far as is needed to supply a convenient algebraic framework for the dis­ cussion of Fary's modification of the spectral sequence of a map (XI). It is obtained by combining Leray1s filtration with a filtration stemming from a decreasing sequence of closed subspaces. To Chapter XVI we have relegated some remarks about the spectral sequence of a map, Including a 1

2

INTRODUCTION

brief summary of its definition, a procedure for putting on it a module structure over the cohomology of the base, a discussion of families of supports, a sufficient condition under which the stalk of the Leray sheaf in a fibre bundle is the cohomology of the fibre, and an application to the cohomology of certain products. As to Chapters I and II, the undersigned would like to point out that cohomology manifolds are emphasized for technical convenience rather than for the sake of generality. "When it comes to manifolds, genuine ones are of the course the most important ones, and differentiable transformation groups already form a very interesting special case. How­ ever, if one does not wish to assume that all group actions (including those on the one point corapactification of a euclidean space) are differ­ entiable, then assumptions of a cohomological character are much more tractable with the methods used in this seminar and have a more "hereditary11 character (e.g., Theorem 4.10 (b) of I, or 3*2 of V are false for mani­ folds ). The discussion of transformation groups starts with Chapter III, where Floyd gives a sheaf-theoretic interpretation of Smiths theory of prime period maps and applies the transfer homomorphism to thediscussion of the orbit space of a finite group. Prime period maps are also studied in Chapters IV and V, to­ gether with the action of the circle group T1, from a different point of view. It consists of a systematic useof the twisted product Xq = X Xq -^j of X with a universal bundle for G. This space on onehandis a fibre bundle over the classifying space of G, with fibre X, and on the other hand it has a projection on X/G, such that the inverse Image of a point Y of X/G can be identified with the classifying space of the stability group of any point in this orbit. It allows us to tie together the cohomology groups of X, X/G, and the fixed point set F, with those of the classifying spaces of the stability groups and of G. After a few general remarks about X^, Chapter IV proves various, mostly known, results about prime period maps on homology spheres and their analogues for the circle group. Chapter V is devoted to the local theorems for these two cases, stating mainly that F and X/G are cohomologically locally connected If X is, and that F Is a cohomology manifold if X is. The discussion of elementary abelian p-groups (by which we mean direct products of cyclic groups of order p If p is prime, of circle groups if p = o ) is resumed in Chapters XII, XIII, with the help of Fary’s spectral sequence. In Chapter XII, it Is shown that if X is totally non-homologous to zero in X^, then G has fixed points. In

INTRODUCTION

3

Chapter XIII a relation is obtained between the dimensions of X, F, and the fixed point sets of the subgroups of index p if p is prime, or of the closed connected subgroups of codimension 1 if p = o, when X is a cohomology sphere or a cohomology manifold. In contrast with the re­ sults of Chapters IV, V, those of Chapters XII, XIII do not seem to follow by induction from the case G = Zp, T1. Chapters V to IX are devoted to some known basic theorems: the local finiteness of the number of orbit types in a cohomology manifold, first when G is a torus (VI), and then in the general case (VII); the existence of equivariant embeddings in euclidean spaces and of a slice (VIII); various results concerning the orbits of the highest dimension in a cohomology manifold (IX). In Chapter XIV it Is shown that if a compact Lie group G acts effectively on a sphere with one class of orbits and is not transitive, then G = T1, SU(2) and G acts freely, and that if G acts on euclidean space with two classes of orbits, then it has a fixed point. Chapter XV presents new results of Bredon pertaining to the action of a compact Lie group acting on a cohomology n-manifold when the fixed point set F has the greatest possible dimension allowed by the results of Chapter IX, namely n - k - 1 where k is the highest dimension of the orbits. The highest dimensional orbits are then in­ tegral cohomology spheres, the orbit space around a fixed point x is a cohomology manifold with boundary P and there is a local cross-section around x for the orbits. In short, these Notes give an exposition of some basic theorems and of results obtained by cohomological methods in the theory of compact Lie groups of transformations. They make no claim at completeness, al­ though it has been tried to give a fairly comprehensive discussion of the topics chosen, and do not cover all aspects of the theory of transforma­ tion groups. It should also be pointed out that purely cohomological methods, while forming a major part of the subject at present, have their limitations, as is shown by well known counter examples; and that in view of this, it would certainly be very desirable to make more effective use of differentiability assumptions than has been possible so far. As to references, each chapter carries its own bibliography. The numbering of lemmas, theorems, remarks, etc. in each chapter is cumulative. Por example 2.1 means Section 2.1 In the same chapter; (IV, 2.1) means Section 2.1 in Chapter IV; 3*2(1 ) refers to formula (1 ) or assertion (1 ) in 3*2, as the case may be. Finally, I would like to thank G. Bredon for his help in checking the final version of these Notes. The Institute for Advanced Study

Armand Borel

Is

COHOMOLOGY MANIFOLDS A. Borel § 1 . Preliminaries

1 .1 . L will always denote a principal ideal domain, ring of integers, Kp a field of characteristic p (p prime or zero), Zp the prime field of characteristic p. All topological spaces will be Hausdorff and, unless otherwise stated, locally compact. A denotes the closure of a subset A of a topological space.

H^(X; L), (respectively H^X; L)) is thei-th cohomology group of X, in the sense of [4], with coefficients in L, and compact (respectively closed) supports. H*(X; L) (respectively H*(X; L)) is 1 1 the direct sum of the groups H£(X; L) (respectively H (X; L)). Thus H*(X; L) (respectively H*(X; L)) is the Alexander-Spanier cohomology ring of X, coefficients in L, with compact (respectively closed) supports (respectively when X is paracompact). write

X ^ Y X ~p Y.

will mean H*(X;

L) = H*(Y; L).

If L = Kp,

we also

Given an open subset U of X the standard homomorphism H*(U; L) -- > H*(X; L) will be denoted by j-^j and its restriction to H^(U; L) by j^j. have trivially (1)

If

V

is open in

U,

H^(X; L) = ind lim (H^(U; L), j^jy)

then

(U, V

= j-^

j^.

We

open relatively compact).

If A is a closed subspace of X, the natural homomorphism H*(X; L) -- > H*(A; L) (respectively H*(X; L) -- > H*(A; L)) is denoted by (respectively r*xa ) and its restriction to elements of degree i by r ^ (respectively p qx A^’ recall the exact cohomology sequence i1 ri (2) -- > h£(Uj L) -£¥-> H^(X; L) -22^_> H£(A; L) -- > H^+1(U; L) -- >

5

Z

the

6 for

CHAPTER I

BOREL U

open in

X

and

A = X - U.

Finally the homomorphism H (Y; L) -- > H (X; L) (respectively * * H (Y; L) -- > H (X; L)) induced by a continuous (respectively and proper) map f : X > Y is denoted by f (respectively f*c )* 1.2. Dimension. By dirn^(X) is meant the value on X of the dimension function function over over LL introduced introduced by by H. Cohen H. Cohen [3], namely [3], namely (3)

dimLX dimLX £< nn hh£+1 £+1(U (U; j L) L) == 00 for for all all open open UU(( XX ..

If L = Kp, we write dim^X for dim^X. The condition on the right hand surJective for all closed subsets A of side is equivalent to: p q x a X (cf. [[ 33]] ,, it it is is proved proved there there for for XX compact, compact, but but the the proof proofextends extends easily to the the locally locally compact compact case). case).ItIt also also implies implies that that H^(U; H^(U; L) == H^(X - U; L) and U U open open in in X. X. Also Alsodim^X dim^X L) == 00 for for all all jj ^^ nn ++ 11 and is ^ the dimension over L of any locally compact subspace, and is the upper limit of the dimension over L of the compact subspaces of X. H. Cartan [2] has introduced the following notion of dimension in ®-cohomology where © is a "paracompactifying family" of closed sub­ sets (see §5): (4) dim d i m#^ jX LX < < = >

Hn+1 Hn+1(U; (U; A) A) == 00 (U (U open open in in X, X, AA sheaf sheaf of of L-modules). L-modules).

This implies H^dJ; A) A) == oo for for jj ^^ nn ++ i. i. By By results results of of [ [^] it-] and and of dimension functions functions are are equal. equal* Floyd (see §5) these two dimension

of

1 .3 . Cohomological local connectedness or k - clcT neighborhood

X X is kk - clc clc over over LL atat x x ififgiven given aa neighborhood neighborhood UU of of xx there there exists exists aa k ==00 (if V of xx inside inside UU such suchthat that Im Im r-gy r-gy (if kk == 00,, one one

has to take reduced cohomology cohomologygroups). groups).

The The space space is is clc£ clc£

(at (at x) x)

ifif

it is k k-- clcL clcL for for all all k < n, °1cl k < n, clcL °1cl for a11 clcLn' forand a11 n'and clc^ it is is clc£ clc£ and and if ifgiven given UU there there exists exists VV such such that that clc-^ ififit Im r = 0. 0. The The space space XX is k - clc^, °1cl > c 1 cl or or clcL clcL ^ ^ ^ is so atatevery every point. point. The The property property clc£ clc£ 3-s3-sequivalent equivalenttotolocal local connect­ connect­ edness. If L = Kp, we replace replace the the subscript subscript LL byby p. These definitions are valid for an arbitrary space. If X is locally paracompact at x then, as is well known [4, p. 193]> given c e Ker r-gx, there exists a neighborhood V of x inin UU such such that Therefore,in in this if and and only only c e Ker i*uy Therefore, this case, case,XX is is kk - clc at x if given a neighborhood UU ofof x x there there exists exists aa neighborhood neighborhood VV of of xx in in U such that Im r^y is is aafinitely finitely generated generated L-module. L-module. It It is is clc clc if if there exists V such that that Im Im r^y r^y is is finitely finitely generated. generated. "We recall that if

X

is compact and

clc

over

L,

then

CHAPTER I:

§2

COHOMOLOGY MANIFOLDS

7

H^(X; L) is a finitely generated L-module for each integer i. This is proved in Wilder [6]. For more recent proofs, see Floyd, Trans. A. M. S. 81*0957), 319-337, E. Dyer, Annals of Math., 67(1958), 119-149. Let Y be another space. The projection of X x Y onto one of the factors induces an injective map in cohomology with closed supports. This and the Runneth rule applied to compact subsets of X and Y show at once that X x Y has any one of the above properties at (x, y) if and only if X and Y have this property at x and yrespectively. In the sequel we deal almost exclusively with spaces of finite cohomological dimension. In that case, the properties clc and clc00 are of course equivalent. §2.

Local Betti Numbers Around a Point

2.1. DEFINITION. The space X has the i-th local Betti number k around x (over L), in writing pi (x; L) = k, if given an open neighborhood U of x,there exist open neighborhoods W ( V ( U of x with W C V, V C U, such that for any open neighborhood W 1 C W of x, Im jy^i = Imjy^- and is a free L-module of rank k. We write pi(x; L) < ai(pi(x; L) is at most increasingly Infinite) if given U there exists V ( U such that Im j^y is finitely generated. The local Betti numbers around a point (a notion introduced by Alexandroff (Ann. of Math. 3.60 935), 1-35)) are traditionally defined with respect to fields of coefficients, and in this case our definition is equivalent to the usual one. When L is not a field, 2.1 will cover our needs, but will probably not be the final definition. It is likely that one will also have to introduce local torsion coefficients, which in our case would be zero. The condition Im jy^, = Im j-^- is not necessary to state a meaningful definition, but seems to be needed wherever one wants to put this notion to use. It would also be desirable to introduce local groups at a point. This is easily done in homology, where these groups form in fact a sheaf in a natural fashion (see [1] and Chapter II). One may of course consider the projective limit of the groups H^(U; L) I 1 with respect to the maps jy^. If L is a field, and if p (x; L) < cd, the dimension of this limit is in fact equal to p^(x; L). When L is not a field, these groups have been introduced and studied by Conner and Floyd (Michigan M. J. 6(1 959), 33-43) under the assumption Im j^, = Im j^made in 2.1 and one further axiom expressing an analogous stability of kernels. 2.2. THEOREM. Let X be finite dimensional over L. Then the three following conditions are equivalent.

8

CHAPTER I

BOKEL (i)

1313

X

and all

subspaces of Im j p Q

(ill)

X

For any pair

with

P

compact,

T his t h e o r e m is d u e to W i l d e r 6 A

in

all

Q

of open

contained in

Q,

[6] w h e n

TTHHEEOORREEMM..

LLeett X X

ddiimm^^XX

H^+ 1 (Vi n V2 )

hJ o j ,

u u2

)

^ + 1 (u, n U2 )

.

The lemma follows then by simple diagram chasing from the nullity of and Im g .

Im f

REMARK. This proof is almost identical to that of Property 6.2 in [1]. The difference is that the condition of being zero imposed on images by Ak L and on pk (x; L) must be replaced by "being finitely generated". One would get similar results in the sameway by replacing "zero" in those conditions by "congruent zero mod C" where C is a class of L-modules in the sense of Serre. § 3*

The Notion of Cohomology Manifold

3.1. DEFINITION. Let n be a positive integer, and L a principal ideal domain. The space X is a Wilder n-manifold if dim^X is finite and if p^(x; L) = 0, pn (x; L) = 1 (i =[ n; x g X). L,

Let X be a Wilder n-manifold. Then by 2.2it is clc over in particular it is locally connected, and by 2 . 3 we have dirn^X = n.

3.2. DEFINITION. A connected Wilder n-manifold is orientable if H^(X; L) contains a free L-module A of rank 1 such that x e X has a fundamental system of connected open neighborhoods U for which A = Im j^j. A Wilder n-manifold is orientable if all of its connected components are orientable. It is locally orientable if each point has an open orientable neighborhood. Thus, if X is orientable, and connected, the intersection of all the images Im j^j, where U runs through the non-empty open subsets of X, is a free submodule of rank 1. It can be seen without diffi­ culty and will in any case follow from 4.3 that in a locally orientable Wilder n-manifold, each point has a fundamental system of orientable open neighborhoods. It is not known whether a Wilder n-manifold is always locally orientable. In this seminar we shall always assume local orientability, and in order to abbreviate, we introduce the following: over

L

3.3DEFINITION. A cohomology n-manifold over L (an n - cm or an n - crn^) is a locally orientable Wilder n-manifold over L.

CHAPTER I

BOREL

REMARK. Wilder n-manifolds over fields are the main topic of [6], which sums up all results concerning them known at that time. Smith, in his work on periodic transformations, has introduced a notion of homology manifold over a prime field Z^, which has turned out to be equivalent to that of an n - cm over Z^ (see Yang [7])• In [7 Yang introduces and studies homology manifolds over the reals mod 1, which correspond to our n - cm over Z. He proves there (in the language of homology) the results 4.2 to 4.5 below, as well as a lemma similar to 2.4, although stated in a more special case.

],

3.4. Remark on the orientability of Wilder Manifolds. Let X be a Wilder n-manifold over the principal ideal domain L, and let x € X. The definition of local Betti numbers shows the existence of a subgroup H(x)C H^(X; L) and of a basic system Ux of open neighborhoods of x for which Im = H(x), Im = o (i 4 n), for U e Ux. With this notation, we have the following assertion: Let X be a Wilder n manifold over L. Then X is orientable if and only if the following is true: H(x) = L for every x € X and if H(y) C H(x), then H(y) = H(x), (y, x e X). The necessity of the condition follows from the definition of orientability. Assume now this condition to be true. Let H be a sub­ group of H^(X; H^(X; L) L) isomorphic isomorphicto to LL and and xx€ X € such X such thatthat H(x) H(x) = H. If U € Ux and and y ye eU,U, then then H(y) H(y)C CH(x), H(x), hence H(y) = H(x) = H. Therefore X(H) X(H) = Cx, H(x) = H) is open in X. Orientability follows now sinceX is locally locally connected connected andandX(H) X(H) n X(H') is empty if H 4 H1. If L is a field, then the condition may be stated slightly more simply in the following way: Let X be a Wilder n-manifold over a field L. Then X is orientable if and only for every non-empty open subset U the homomorphism has a non-zero image. We also remark that the condition MIm empty open UM is equivalent to MIm 4 0 and that above, we may also define H(x) as Im §4.

4 0 for all non­ aH non-empty open UM, e Ux )*

Some Properties of Cohomology Manifolds

4.1. We recall first the so called "minimality principle" for (Alexander-Spanier) cohomology with compact carriers: Given c € H^(X; L),

CHAPTER I:

COHOMOLOGY MANIFOLDS

c 4 o, the set of closed subspaces for which inclusion, has a minimal element.

*#•

**cXF(c) 4 °>

ordered by-

PROOF. Let be a totally ordered set of closed subsets on which c induces a non-zero element, and let F = nj_ejFi. We want to show that r £p(c) 4 °* Assume it is not true, and let K be a fine cover (couverture fine), in the sense ofLeray, for cohomology with compact carriers. Its section by a closed subset A, i.e., the quotient of K by elements with supports in X - A, endowed with the supports, S(c) n A, is then a fine cover for A. Therefore, if * ^ ( 0 ) = °> there exists a representative cocycle c 1 of c in K and k € K such that S(c’ - dk) does not meet F. Since it is compact there exists an i € I ■Xfor which S(c* - dk) n F^ = , whence r^p (c) = o. The minimality principle follows from Zorn!s lemma. ^ 4.2. LEMMA. Let X be an n - cm and F be a closed subspace of X. Assume that for any rela­ tively open non-empty subset C of F, Im 4 °* Then F is open in X. It is enough to show that U n F is open in U when U be­ longs to some opencovering of X; since our assumption on F is a fortiori fulfilled for U n F, we may assume X to be orientable and connected. Let Y bean open subset of X meeting F such that ^ J’xy = nv Im (Vrunning through the non-empty open subsets of X). We want to prove that Y ( F. We consider the following commutative dia­ gram where the horizontal lines are parts of cohomology sequences 1 (2 ), and the coefficients L are omitted. h £(X

- F)

->

h £(x

A

>

h £(f

)—

> 0

A

JXY h£ ( y

)—

in JF,YfiF

- y n f ) — > h £ ( y ) — > h£ ( y n f ) — > 0

By 2.4, H^+1(Y) = H^+1 (X) = 0, hence the horizontal lines are exact. If Y n F 4 Y then by our choice of Y, Ira = Im y-YHF^ nullity of Im jp YnF follows then by diagram chasing, whence a contradiction. 4 .3 . Then

THEOREM.

Let

X

be a connected

n - cm

(1 ) For every non-empty open subset

U,

over the

L.

12

BOREL

CHAPTER I

homomorphism is surjective, hence H^(A; L) = o for every proper closed sub­ set A of X. (2 ) X is orientable if and only if H^(X; L) = L. If X is orientable and U is an open sub­ set, then U is orientable and, if U is moreover connected, is an isomorphism. If X is non-orientable and L is a field, then H?(Xj L) = 0. If is not surjective, there exists a c € H^(X; L) and a proper closed subset A for which rcxA^c ^ ^ °* By 4.1 we can find a minimal proper closed subset F such that ^ x f ^ 4 °* Consider the commutative diagram, where the two columns are exact by 2 . 3 and where L is omitted.

HS(U)

--- > Hc(X)

J/ h£ ( u

J/

n p ) — > h £ ( f ) — > h£ ( f - f n u )

si 0

P ■X-

the maps being either restriction maps r or inclusion maps j . If * F n U is non-empty, then rQX F_Fnu(c) = 0 by the minimality of F, hence rcxp^°^ belongs to the image of Fnu, and F is open by 4.2. Since it is closed, non-empty, proper and X is connected, this is a con­ tradiction. The second part of (1 ) follows then from the first part by the cohomology sequence and 2 .3 , and (2 ) Is an obvious consequence of (1 ) and the definitions. 4.4. DEFINITION. Let X be an n - crn^, and U be an open connected orientable subspace of X. An open connected subset V of U is said to be adapted to U if is an isomorphism and = 0 for i/n. By 4.3, each point x e U hoods which are adapted to U.

has a fundamental system of neighbor­

4*5* COROLLARY. Let X be an (orientable) n - cm over Z. Then X Is an (orientable) n - cm over any principal ideal domain L. This follows from 2 . 4 , 4 . 2 and the universal coefficient formula

CHAPTER I:

§1+ 0

>

h £(X;

Z) ® L -- >

13

COHOMOLOGY MANIFOLDS

h £(X; L )

-- > Tor(h £+1 (X; Z), L ) -- > 0

4.6. COROLLARY. Let X be a connected n - cm and F be a proper closed subset. If F is an m - cm, then m < n. Let X be a point of F which is not an interior point. Then F n U is a proper closed subset of U for any open connected U con­ taining x, and therefore H^(F n U; L) = o by 4 .3 . Thusthe n-th local Betti number of F around x must then a fortiori bezero, and 4.6 follows from the definitions and 2.3* 4.7* COROLLARY. Let X be a connected n - cm and A be a closed subset such that A < n - 2. Then X - A is connected. It is enough to show thatif U is a connected open neighbor­ hood of an arbitrary point a € A, then U - U n A is connected. Hence we may assume X to be orientable. Since, by the cohomology sequence, H^(X -A; L) = Hc(X; L), our assertion follows from 4 .3 . 4.8. PROPOSITION. Let X be a connected n - cm over L. If L = Z2 then X is orientable. If L = Z and X is not orientable, then H^(X; Z) = Z2, hence X considered as an n - cm over a field of characteristic 4 2 n°t orientable. Assume L = Z2. Since X is locally connected, Z2 ) is the inductive limit of the vector spaces H^(U; Z2 ), where U runs through the relatively compact open connected subsets of X. Each one of those is contained in a finite union of connected orientable open sub­ sets. Therefore It will be enough to show that if U, V are open connected orientable, then U U V is orientable, i.e., h £(U U V; Zg) = Zg. To this end we consider the Mayer-Vietoris sequence (1 )

h£(u n v) —

> h£(u) + h£(v) -£-> h£(u u v) —

> o

.

For each connected component W of U n V, j*j^ and j^y are isomorphisms "by 4.3, hence the image of f is the subgroup generated by (a, b) where a and b are the generators of H^(U; Z2 ) and H^(V; Z2 ) respectively. Therefore Im g is 1-dimensional. Let now L = Z. From 4.3 we know that if X is not orientable, H^(X; Z) is a proper quotient of Z, and since X is an n - cm mod 2, it follows from what has just been proved that H^(X, Z) is not zero, of finite order divisible by 2 . It remains to show that its order is -> dim^X dim^X< h£(u) —

"Is

T

T

> h £“1 (v n f ) —

> h £(v - f ) —

> h £(v ) —

T h £"1 (v)

COHOMOLOGY MANIFOLDS > o

> o

where the rows are parts of exact cohomology sequences, the vertical maps are j maps and cohomology is with respect to L. By 4 .3 , H^(U) = H^(V) = L,is an isomorphism, and H^(V - F) is a free L-module whose rank is equal to the number of connected components of V - F. Therefore, 6 is not zero if andonly if V - F is not connected. More­ over, since j ^ 1 = 0 , it followsthat if j ^ p vnF 4 0, then V - F is not connected. Let now dim^F = n - 1. Then F has no interior point by (a). Also, by 2 . 3 there is an x e F such that the n-1-th local Betti number of F at x is not zero.This, together with the fact that in X, pn”1(x) = 0, implies that we canfind an open connected orientable U containing x, and abasic system of open connected orientable neighbor­ hoods V of x in U such that j ^ 1 = 0 and the remark just made, V - F is not connected.

jjjnF VflF ^ °*

Then

If now F separates X locally around x, then for suitable V, 5 is not zero, hence H^”1(V n F) ^ 0 and dim^F ^ n - 1; if, more­ over, F has no interior point, then dim^F = n - 1 by (a). 4.10*THEOREM. (a) Let X be an s - cm and Y be a t - cm over L . Then X x Y is an (s + t ) cm over L, and is orientable if and only if X and Y are orientable. (b) Let L= Z, Kp. Let X x Y be a connected n - cm^. Then X is an s - crn^ and Y a t - cm^ (s + t = n), both orientable if and only if X x Y is orientable. The first part is an elementary consequence of the definitions and the Kunneth rule, and is left to the reader. We use implicitly the fact that dirn^ ( X x Y ) is finite. More generally, it is known that din^ (X x Y) < dim^ X + din^ Y for arbitrary locally compact spaces X, Y. This follows for instance from Proposition 4 in [2, XXI]. Let now X x Y be an n - cm^: Then (see end of 1.3 ) X and Y are clc^. Let first L = Kp. Then, for subspaces U and V of X and Y respectively, we have H£(U x V; L) = E(*(Uj L) (x) H*(V; L) and this isomorphism commutes with maps. Our assertion follows in this case easily from 4.3. Let now

L = Z.

Then

XxY

is an orientable

n - cm

over

16

CHAPTER I

BOREL

if it is orientable over Z (4.5). Therefore, for each p (prime or zero) integers s , t Is an Sp - cm and Y a tp - cm mod p, both Let U C X, V C Y be open connected, such that the Kunneth theorem, there is an exact sequence (1) 0 —» EaH^(U;Z) ® H £ " a (V;Z)-» h£(X

Y;Z)-*

x

Since the middle term is isomorphic to exists an integer s(U, V) such that

^ ( U ; Z ) 0 ^ ‘a(V;Z) = 0 (a

4 s(U,V)),

by the above, there exists (s + t = n) such that X orientable If X x Y is. U x V is orientable. By

Tor(H^+1 (U;Z),H^"a (V;Z)) ^

Z by 4.3, it follows that there

H^(U;Z) ® H^“a(V;Z) = Z (a = s(U,V)) .

Let U ! C U, V 1 C V be open and connected. The exact sequence (1 ) being natural with respect to maps, 4 .3 (2 ) shows that s(U*, V 1) = s(U, V). Since X x Y is connected, it follows at once thats(U, V) is inde­ pendent of U, V. Call it s and put t = n - s.Thus wehave H^(U; Z) ®

h £'s (V;

Z) = Z

whenever U C X, V ( Y are open, connected, and such that U x V is orientable. For U ! CU, V 1 C V open connected and small enough, U ! x V 1 is adapted to U x V (4.4) and Im J'v'V are generated (2 .2 ). Applying 4 . 3 to the case L = KQ we see that these groups, modulo torsion, have rank one. Therefore, if Un C U 1 and V M C V 1 are open connected, Im yii and Im jy yu have finite indices in Im and Im jy y t respectively. However, we have already proved that the composition of the maps

Im

Im J’u,u'

®

Im JV,V'

Hc^U;

® Hc H*(W; Z ) ® Z p has at least dimension two; the -universal coefficient formula and naturality * imply then that Im j^ z has dimension ^ 2. This, being valid when W, W f describe basic systems of neighborhoods of x, shows that the sum of the local Betti numbers mod p of M at x is at least two, in con­ tradiction with the fact that Mis a cm over Zp. Therefore S = (e) and we have shown the existence of a subgroup B(x) = Z of H (U; L) * which is contained in Im jy y z for every open neighborhood V of x in U, and is equal to this image if V is small enough. Let n(x) be the integer such that B(x) C H^ (U; Z). From the universal coefficient formula, n(x) = np, for allp, thus n = n(x) is independent of x, and all np are equal to n. This shows already that M is a Wilder n-manifold over Z (3-1 )• By 3*4, local orientability will follow if we prove that B(y) C B(x) (y, x € U) implies B(y) = B(x). Suppose it is not so, and let p be a divisor of the Index of B(y) in B(x). Then we would have Im jjj ^ ^ = 0 f°r a sufficiently small open neighborhood W

of

y;

however,

U

being orientable

mod p,

this contradicts 4 .3 .

REMARK. An extension of 4.10 to generalized manifolds with boundary will be found in a forthcoming paper of F. Raymond, "Separation and uniqueness theorems for generalized manifolds with boundary" to appear

18

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CHAPTER I

in Michigan J. Math. § 5•

Appendix on Cohomological Dimension

This section is mainly devoted to the proof of 5«3* This theo­ rem is due to Grothendleck for the case L = Z, X paracompact [5, §3- 6a; P« 197], and to Floyd (unpublished) in the general case. 5.1. LEMMA (Floyd). Let S be a class of sheaves on the (not necessarily locally compact) space X satisfying the following conditions: (i) If

A €S

and

A 1 = A,

then

A* € S.

(ii) If o -» A" -> A A 1 -> o is exact, then A, Ar< e S implies A 1 e S and A 1, A n € S im­ plies A € S. (iii) If A = U^A^ where (A^) is a directed upward family of subsheaves of A belonging to S, then A e S. (iv) For each open U( X, the sub sheaf Ly = (U x L) U (X x o ) of the constant sheaf L = X x L is in S. Then PROOF.

© ielkj.

S

consists of all sheaves on X.

Every sheaf A is a homomorphic image of a direct sum 1^2 ]; here i ranges over all sections A and the

homomorphism cp is defined naturally from the sections. If cp(©LieI,Lu ) e S for each finite family I1 C I, then A € Sby-{iii). Hence It is1enougfr toprove that A e S whenI is finite. By (ii), it is then sufficient to prove that every homomorphic Image A of a sheaf Ly belongs to S; by (ii) and (Iv) it is therefore enough to show that every subsheaf of L belongs to S. L, being a principal ideal domain, is a unique factorization domain. We denote b y N(I), (I proper ideal In L), the number of prime ideals of which I is the product. Thus I' 3 I, implies n(I* ) < n(I). The stalk of A at x € X Is an Ideal of L, and we have clearly Ay } A^. for y sufficiently near x; therefore if is the subspace of points x for which n (Ax ) < then the form an in­ creasing sequence of open subsets whose union is U = {x, A^. 4 and A is the union of the increasing sequence of subsheaves A^_ By (iii)> it is enough to show that the

A^

belong to

S.

We have

AQ = Ly

€ S,

CHAPTER I:

§5

COHOMOLOGY MANIFOLDS

19

hence, using (ii) and induction on i, it suffices to show that e S. This last sheaf is concentrated on - U.^; moreover - U^_1 is the union of disjoint relatively open subsets V. where j runs through the ideals !• of L which are products of exactly i prime factors and where is equal to I. on V-. Since I. = L as an L-module, it J J J follows that there exist open sets W., W*. with V. = W. - Wl such that J

V. = J

J

J

J

J

(iv), (ii) and (iii) imply then that

J

J

* € s#

5.2. A paracompactifying family © on a (not necessarily locally compact) space X is a family of closed subsets of X satisfying the following conditions: (a) (b) (c) (d) (e)

The elements of $ are closed and paracompact. $ is closed under finite unions. Every closed subset of an element of $ belongs to Every element of $ has a neighborhood in $. The elements of ® cover X.

Except for'(e), which is added here for convenience, these are the usual conditions of H. Cartan [2, Exp. XVII; b p. 150]. When X is locally compact, as again will be tacitly assumed from now on, any $ con­ tains the family of compact subsets of X. 5«3« THEOREM. Let © be a paracompactifying family on X. Then, in the notation of 1.2, we have dim-j-X = dim^ ^X. PROOF. (1 )

For

U

open in h £(U;

X,

we have

L) = H£(X; L^)

where the left hand side means the cohomology of U with respect to the family of elements of ® contained in U [4,p. 1 9 0 ]• If moreover U is relatively compact, then f |U is the family of compact subsets of U, hence (2)

L) = l£(Uj L)

(U

compact).

Therefore, by 1(1 ), dimLX < dim$ LX. We now prove that dim^X ^ dim^ jX when \jr is the family of compact subsets of X. We have then to show that if H^(U; L) = 0 for each open U (for a given n), then the class S of sheaves for which H^(X; A) = 0 (j ^ n) consists of all sheaves, and

20

CHAPTER I

BOREL

for this it is enough to show that S satisfies the conditions (i), (ii), (iii), (iv) of 5*1•(i) is clear, (ii) followsfrom the exact cohomology sequence [4, p. 174], (iii) from the equality h|(X;

[4, p. 19M* in X).

A) = ind lim H;j(X; Aj_)

and (iv)from the equality

h £(U;

L) =

H^(X; L jj)>

(U open

In case X is paracompact and © Is the family of closed sub­ sets of X,then dim^^X = dirn^^X is proved in [4, p. 1 9 6 ], (for L = Z, but the proof given there is seen also to hold for a general L). Consider finally the general case. [2, XVII p. 6], we have

By a lemma of H. Cartan

d i % , L X = max ditn«|M,LM

where M runs through the elements of $ and $ | M consists ofthe elementsof a> contained in M, thatis, of allclosed subsets of M. Hence dlm ®,LX = max dim ® |M,LM = max din V ,L M = d il% ,L X = dimLX

5.4.

COROLLARY.

on X.

Let

$ be a paracompact ifying family

Then diti^X < ni f and only

for all open U

*

if

h£+1

(U; L) =0

in X.

1(1) and 5-3(1 )> (2 ) show that the condition (V i } : ^ +1(u; L) = 0 for all open 5.3(1 )•

U

implies

dim^X < n.

The converse follows from 5*3and

A particular case: Let X be paracompact and $ /be the family of closed subsets. Then H^(U; L) may be identified with the relative Cech or Alexander-Spanier cohomologygroup H^(X mod A; L), where A = X - U. Therefore dim-jOC £ n Ifand only if Hn+1 (X mod A; L) = 0 for every closed subset A.

§5

CHAPTER I:

COHOMOLOGY MANIFOLDS

BIBLIOGRAPHY [1]

A. Borel, "The Poincare duality in generalized manifolds," Mich. Math. J. 4(1957), pp. 227-239-

[2]

H. Cartan, Sem. E. N. S. 1950-51.

[3]

H. Cohen, "A cohomological definition of dimension for locally compact Hausdorff spaces," Duke M. J. 22(1954), 209-224.

[4]

R. Godement, Theorie des faisceaux, Act. Sci. Inc. 1252, Paris 1958, Hermann ed.

[5]

A. Grothendieck, "Sur quelques points d !algebre homologique," Tohoku Math. J. 9(1957), PP- 119-221.

[6]

R. L. Wilder, Topology of manifolds, Amer. M. S. Coll. Publ. 32 (1 949).

[7]

C. T. Yang, "Transformation groups on a homological manifold," Trans. A. M. S. 8 7 (1 9 5 8 ), pp. 2 6 1 -2 8 3 .

21

II:

HOMOLOGY AND DUALITY IN GENERALIZED MANIFOLDS A. Borel

Our main purpose here is to obtain a Poincare duality theorem in cohomology manifolds. Even if one is interested mainly in a state­ ment involving only cohomology, one has to use in the proof groups which play the role of homology groups, and therefore this presupposes some homology theory. The approach of [2] led to satisfactory results only when the coefficients form a field. We shall give here another one, which will allow us to make use of universal coefficient formulas in the general case. The present paper, which differs considerably from the oral lecture, represents joint work of J. C. Moore and the author. The notation is that of I. We assume familiarity with sheaf theory [3, 5]* In particular the following notion and notation of [5] will be used: A family of supports in the space X is a collection of closed subsets such that the union of two elements of $ belongs to o, and that every closed subset of an element of $ is in ©«» Given a sheaf F on X, r(F), r$(F), F(A) (AC X) denote respectively the sections of F, those with support in $, and the sections over the subspace A. "Section" will always mean "continuous section". A sheaf F is called flabby (respectively soft) if the restriction map r(F)-> F(A) is surjective for every open (respectively closed) subspace A of X. Thus flabby (respectively soft) sheaf is the translation of faisceau flasque (respectively mou) of [5]• If K is a grating on X, Kg will denote the subgrating of elements with supports in the subspace U. L is a principal ideal domain given once for all and Horn, Ext will be taken over L. Unless otherwise stated, all topological spaces are locally compact and Hausdorff. § 1.

Homology groups for locally compact spaces

1.1. DEFINITION. A sheaf C of L-modules on X is a funda­ mental sheaf for homology with coefficients in L (and supports in a given family $ of supports) if 23

BOREL

2b

(i) _C by

CHAPTER II

is graded, and has a boundary operator lowering degrees

1 .•

(ii) C_ is flabby (or $-soft, or ®-fine, if compactifying family, see I, §5)* (iii) (1 )) (1

For each open

U

in

X

©

is a para-

we have a split exact sequence

0 O -» -> E Ext(H^+1 x t(H g+1 (U; (U; L L), ), L L)) -» H H ^^CCddJJ)))) -> > Hom(Hg(U; Hotn(H^(U; L L), ), L L)) -» 0

and if V is open in In the maps induced from

U, the sequences (1) for U, V are compatible with j-yy and the restriction C(U) -> C(V).

If L is a field, and X is paracompact, then for any funda­ mental grating for the cohomology with compact carriers K, for instance the Alexander-Spanier cochains with compact supports, the sheaf associated to the Hom(K, L) L) is is fundamental fundamental for for homology homology [2]. [2], thegrating grating Hom(K, 1 .2. Let Let CC- be be aa of supports on X. Then for morphism H*(C(U))-> H*(C(V)) its restriction to H^(C(U)).

fundamental fundamental L-sheaf, L-sheaf, and and let let ®® be be aa family family U } V open we denote by the homo­ induced induced by bythe therestriction, restriction,and andbyby We We put put

H±(X; L) = H j_(r(C )); H ?(X> L) =

L) =

* X)

(2 )

H*(X; L) = £K,_(X; L );

h£(X ;

L) = 2H?(X; £H?(X; L ); H^(X; L) = ZH*(X; L )

.

Here CxCx denotes denotesofofcourse coursethe thestalk stalk ofof C C atat x;x; the the group group H^(Cx ) is the i-th i-th local local homology homology group group at at x, x, and, and, in in case case say say ititisis finitely generated, the rank of H^(CX ) modulo its torsion subgroup corresponds toAlexandroff1s Alexandroff1slocal local Betti Betti number number in in xx (see (see [2] for some some remarks on the relation relationbetween betweenthe thelocal localBetti Betti numbers numbers in in x and those those around x, as defined in I). The The groups groups H?(X; H?(X; L) L) are are then the stalks stalks of the homology sheaf Hj_(C), and and H(C) H(C) will will be be called called the local homology homology sheaf. By 1.1(iii), H*(C(U)) is independent of the particular funda­ mental sheaf considered, and this legitimizes the notation H*(U; L). We shall allow ourself to use it for the local groups, although we do not know whether the axioms are sufficient to insure uniqueness, because this will be the case anyway for cohomology manifolds. We shall come back to this in 2.5 and 5•1• 1.3* THEOREM. Any locally compact space, has fundamental sheaves for homology with degrees > - 1. For any such

CHAPTER II: HOMOLOGY AND DUALITY

§2 sheaf (3)

C

and any

x eX

25

we have an exact sequence

0-» Dir lim Ext(H^+1(U; L), L) -> Hj_(Cx )-» Dir lim Hom(H^(U; L), L)-» 0 U running through a fundamental system of neighbor­ hoods of x, and the limit being taken with respect * to the maps induced by the homomorphisms

For the proof, see Section 3- Another construction, valid for closed subsets of euclidean spaces, will be given in Section 4. REMARK. The homology groups defined by N. Steenrod (Annals 41 ) by means of regular cycles for a compact metric space X are known to be related to the Cech cohomology by a split exact sequence (1) with U = X) (Eilenberg-MacLane, Annals 43, p. 825)* In this case our groups coincide therefore with the Steenrod groups (except however for a shifting of the degrees by 1 ). Note that with our convention for the degree we have by (1 ), IL^X; L) = 0 for i < - 2, but H_1 (X; L) = Ext(H^(X; L), L) may be nonzero if the space is not locally connected. §2.

Duality in generalized manifolds

2.1. DEFINITION. The space X is an L - n space if dimL (X) is finite and if for any x e X, we have H^(X; L) = 0 (i 4 n) and H ^ ( X;L)=L. An L - n space is locally orientable (respectively orientable) if the local homology sheaf is locally simple (respectively simple). In the locally orientable case, the sheaf L) of the local groups will be called the orientation sheaf. A locally orientable L - n space which is clc^ will also be called a homology n-manifold over L (a n - hm^). 2 .2 . THEOREM.

Let X be an L - n space, _C_ a funda­ mental sheaf for homology over L, and $ a family of supports in X. Then (1 )

H^(X; L) =

•

Let us denote by A the sheaf C with the modified grading A1 = C ^ . Then H1^ ) = 0 (i 4 o), H°(A^) = L, and Hn_1(rt(A)) = Hi(r$(C)). By [5, Chap. II, Theor. 4.6.1] there is a spectral sequence in which

26

BOREL

CHAPTER II

EP’q = 0 (q 4 0), eP’° = HP(X, H° (A^)) = h£(X; H^C).) and is the graded module associated to some filtration of HCr^CA)); the dim^X being finite, this filtration may be assumed to be regular or even defined by a finite number of submodules [5, p. 1 93]• Since we have here E^'^ = o (q =f o), E^'° = E^'0, the theorem follows. REMARK. Let us denote by cp the isomorphism of 2.2 as obtained in the above proof, and let U V be open. Then we have a commutative diagram

V r*nu ^ U )} —

>

Sn(Su)}

*

where r is the restriction map in cohomology, and $ n U, ® n V de­ note the families of intersections of the elements of © by U and V respectively. This follows from the fact that the proof of the basic result of [5] quoted above is obtained by examining the two spectral se­ quences of a double complex, namely the sections of a flabby resolution of C and that the restriction of such a grating on U or V has the same properties with respect to U or V. 2.3* COROLLARY. Let X be an L - n there is a split exact sequence (3)

space. Then

0 —» Ext(f£+1(X; L), L)-> Hn_1(X; 1^(0))-» Hom(^(X; L), L) -» 0 where

C

is a fundamental sheaf for homology on X.

We refer to [2 ] for some consequences of 2 . 3 when L is a field, in particular for the fact that if X is connected orientable, then the isomorphism H^CX; L ) -> H o m d ^ ^ X ; L), L) is induced by the cup product pairing; and for remarks on the relation between 2 . 3 and the results of Wilder [6]. 2.4. PROPOSITION. (a) Let L be afield. Then a Wilder n-manifold over L (respectively a n - cm over L, respectively an orientable n - cm over L) is an (respectively a locally orientable, respectively an orientable) L - n space which is clc over L, and conversely.

C H A P T E R II:

§3 (b).

L et

an (orientable)

(I, 2.2). *

be a n (orientable)

n - cm cm o ov ve er r L. L.

Let

X

be a Wilder n-manifold over

If

L

is a field, t h e n f o r a n y o p e n H^(U;s L )

A s s u m e n o w that

L - n, L

TThheenn

and

L. L.ItIt isis clc clc U, H^(U; H^(U; L) L)

X

is

b e i n g a g a i n a field.

clc clc

oovveerr LL is is the the ddual ual

oovveerr

LL

X

is a n

V

ofofxx iInn U U such such tha thatt

j^y-

n - cm

ove r

Let now L.

U

Im Im jjjyjjjy-

x e X

X

of of

= = 00

TThheerree­­ =and L

U. U.

ffoorr

sati s f i e s c o n d i t i o n

U, U, and and mmoorreeoovveerr that that oovveerr U, U,

i s o m o r p h i c to the the simple simple sheaf sheaf

x. x.

T h e n (I, 4.3) 4.3) H^(U; *^(U; L) L) = L

has n e i g h b o r h o o d s f ofro rwwhhiicchh ImIm

This, t o g e t h e r w i t h 1 . 3 shows tha t p o i n t of

is is

b eb ea accoonnnneecctteedd o r ioerniteanbtlaeb l e

is a n i s o m o r p h i s m f o r e v e r y c o n n e c t e d o p e n s ubset s ubset VV

each point

xx

U n d e r the l a t t e r c o n d i t i o n it is e a s i l y see n that

is a W i l d e r n-mani f o l d .

o p e n subset of a n

L - n

and and s astaitsifsifeise s c ocnodnidtiitoino n

d i m H^(X; L )L ) isisequal equaltotothe thei -it-ht hB eBtettit in unmubmebre rofof X X aarroouunndd X

is is

T h e n f ofro raannyy ooppeenn nneeiigghhbboorrhhoooodd UU ofof

t here e x ists a n o p e n n e i g h b o r h o o d fi n i t e dimens i o n a l . fo r e

X X

(V (V o poepne ni ni n U)U) isisthe thet rtarnasnpsopsoes eof of

T h e n it f o l l o w s w i t h o u t d i f f i c u l t y f r o m 1 .3 that

space.

27

n - hm^.

v e c t o r space to jjjy

X

HOMOLOGY AND DUALITY

and

Also, Also,

ii 44 n. n.

L - n

at e a c h

the the oorriieennttaattiioonn sshheaf eaf HH^^CC))

UU xx H^(U; H^(U; L). L).

is is

This This pprroovveess (b) (b) and and the the

r e m a i n i n g d i r e c t par t of (a). Let now

X

be a n

n - hm-^, hm^,

where

L

is a field, an d

UU bbee aa

c o n n e c t e d o r io er ni te an bt la eb l eo p eo np e nsubset. subset. T h eT nh ebn y b y2.2, 2.2, H ^n X(X; ; LL)) == H°(X, H°(X, L)L) = = L.L. U s i n g the ddiiaaggrraamm ((22),), ffoorr

ii == n, n,

it it ffoolllloowwss that that if if

VV

is o p e n is o p e n

c o n n e c t e d in in U, U,

tthheenn is a n i sis o maonr p ih si os m o. r p hBu i stm .h e rBu e t ^h(eUr;e L) =

Hom(H^(U; L), L ), L)

and

H^(V; L) = L

is t r a n s p o s e of

and

is a n i s o m o r p h i s m .

Therefore Thus

U

^(U;

L) =

H^(U; L) =

is a n o r i e n t a b l e

W i l d e r n-manifold, and this ends the p r o o f of (a). 2 .5 .

Since it is no t c l e a r that the c o n d i t i o n s pu t o n a f u n d a ­

m e n t a l sheaf f o r h o m o l o g y i n sur e u n i q u e n e s s of the d e r i v e d sheaf, t h e r e is some a m b i g u i t y i n 2.1, b e c a u s e it is not said w h e t h e r the c o n d i t i o n L - n

is r e q u i r e d f o r some or f o r al l f u n d a m e n t a l s heaves f o r homol o g y .

The simplest is p r o b a b l y to r e q u i r e it just f o r the r e s t r i c t e d f u n d a m e n t a l sheaves i n the sense of 5*1. the following: if L - n

X

W e a l s o n o t e that the p r o o f of 2.4 shows

is a n (or ientable) n - cm t h e n it is a n (orientable)

space f o r a n y f u n d a m e n t a l sheaf;

(locally orientable, o r i e n t a b l e )

L - n

if

L

is a field, an d

X

t h e n it is f o r all f u n d a m e n t a l sheaves. §3• 3.1. of

E x i s t e n c e of f u n d a m e n t a l s h e a v e s f o r h o m o l o g y LEMMA. X, K

Let

t

h e th e f a m i l y of c o m p a c t s ubsets

be a ^-fine g r a t i n g

supports, and

A

is a n

space f o r some f u n d a m e n t a l sheaf,

[3b, X V I I I - 9] w i t h c o m p a c t

be a n L - m o d u l e .

Let

GC 1

b e the

28

BOREL

CHAPTER II

pr^sheaf over X defined by C ’dJ) = Hom(Ky, A), the restriction map Syy (V open in U) being the transpose to the inclusion K y -- > Ky. Then the natural map of C 1 into the presheaf defined by the cross sections of the sheaf C associated to _C1 is an isomorphism, hence [5, p. ill], C'CU) = C(U). If A is injective, then C is flabby. By [5, Chap. II, 1] our first assertion is equivalent to the following one: Let ^ea family of open subsets of X, and U be their union. Then (a) If t, t 1 e C^U) have equal restriction on each U^, then t = t 1. (b) Letthere be given t^ e 3U°k a ¥aY that for all i, j e l, t^ andt. have equal restrictions on n Uj. Then there exists a t € C !(U) whose restriction to is t^ for every i € I. PROOF of (a): Let c e Ky and I* be afinite subset of I such that the with i e I* cover the support S(c) of c. Let V be an open covering of X formed by set VQ whose closure does not meet S(c) and by elementsV ^ d e I 1) such that V^C U ^ d e I*). Let (r.) be a partition of K subordinated to this covering. Then c = zj_ejirjc and rjC e Ky^. By assumption t(r^c) = t ’(r^c) for all i e I 1, hence t(c) = t !(c)

by linearity.

PROOF of (b): Let c, I1 and V be as before. We tentatively define t by t(c) = £j_€j, t^(r^c). In order to legitimize this, we have to prove that t(c) is independent of the different choices made. Let then J be another finite subset of I such that the U.(j e J) cover J c and let us construct as before endomorphisms s. of K such that J s .(K) C Ky and that c = s - jS -c. We have to show that t*(c) = zt .(s .c) J ^ J

is equal to

J

t(c).

We have

3jt^c e K y L —> Aq —> A1 —> 0

,

hence (2)

Hq (A) = L, H±(A) = 0 (i ^ 1 ) . Let

C = Hom(K, A)

be endowed with the grading

Ci = HotnCK1, A0 ).+ Hom(Ki+1, A1 ) (thus

= 0 i < - 2, C_1 = Hom(K°, A1 )), df(k) = d(f(k)) + (-1 )1+1f(dk)

and the differential (f e C±, k € K)

.

Then U -- > Horn(Eg, A) is the presheaf of cross sections of a flabby sheaf C by 3 .1 . By [4, Cor. 5.2, p. 373 ] or [5, Theor. 5.4.2, p. 1 0 1 ] and (2 ) above, we have then an exact sequence (3 )

0-» Ext(H1+1 (Ky), L) -» H1(Hom(Ku,A)) -»Hom(H1(KIJ),

L) -» 0

.

Moreover, if F = F1 + FQ is a projective resolution of Ky, the sequence (3 ) is canonically isomorphic to the corresponding exact sequence for F "by [^> PP* 373-4], hence it splits, by [4, Theor. 3*2a, p. 114]. Also, this exact sequence is compatible with homomorphisms, therefore C has all the properties required in 1.1. Since C Is flabby, its stalk Cx at x Isequal to the quotient of r(C)= Hom(K, A) by the submodule of sections vanishing at x, and also to . Dir lim (Horn(Kg, A), Syy), U running through the open neighborhoods of x. Therefore

CHAPTER II

BOREL

30

R±(0X ) = Dir lim H^U; L) and the exact sequence 1 . 3 follows then from 1 .1 (iii) and the fact that direct limits and exact sequences commute. 3*3« REMARK. Since A is an injective resolution of L, the homology of Hom(K, A) represents the "hyperhomology invariants" of Hom(K, L) in the sense of [4, Chapt. XVII], which may differ from H(Hom(K, L)). J3ne could also compute them as H(Hom(F, L)), F being a projective resolution of K. § 4.

Closed subsets of euclidean spaces.

In this section we describe briefly another construction of fundamental sheaves for homology valid forseparable metric spaces of finite covering dimension, which makes use of basic gratings. 4.1. A basic L-grating (carapace basique, cf. [3b, XVIII-6]) is a grating, which is a free L-module and has moreovera basis such that the support of a finite linear combination ofthe x^ Is ex­ actly the union of the supports of the x^ which have a non-zeroco­ efficient. For instance the singular chains with coefficients In L always form a basic L-grating. 4.2. LEMMA. Let X be a separable metric space of finite covering dimension. Then (a) X has a fundamental sheaf for cohomology which is homotopically fine and whose sections with compact carriers form a basic grating with degrees bounded from above. (b) X has a fundamental with degrees bounded from above.

L-sheaf for homology

We recall first the operation of section of a grating K on a space X by a subspace Y in the sense of Leray (see e.g., [1, Exp. II]): as amodule it is K/KX_Y, the support of c e K/Kx _y being defined as the intersection of Y by the support of any element of the Kx_y-coset represented by c. This section is clearly fine (respectively homo­ topicallyfine) if K is. Moreover if K is basic with basis (x^), then K/Kx _y is also basic, with a basis represented by the x^ whose support meets Y. Therefore, if X has a homotopically fine basic grating with compact supports, so does every closed subset Y of X. Moreover, a basic grating with compact supports K can always be identified

CHAPTER II:

§5

HOMOLOGY AND DUALITY

31

with the sections with compact supports of its associated sheaf F. Since the associated sheaf of the section of K by Y is clearly the sheaf Fy induced by F on Y, and since Fy is a fundamental sheaf for cohomology on Y if F is on X, we see that if (a) is true for X, it is true for every closed subspace of X. By the Menger-Nobeling theorem, a locally compact separable metric space of finite covering dimension is homeomorphic to a closed subset of some euclidean space Rn . It is therefore enough to prove (a) for Rn . In this case we take the sheaf S of germs of singular chains with coefficients in L, giving the degree n - i to the i-dimensional chains. It is well known to be a fundamental sheaf for cohomology. Its sections with compact supports are then the singular chains with co­ efficients in L. This proves (a). Let K be the grating thus obtained. Its degrees are therefore bounded from above by n. Let C 1 be the presheaf defined by U -- > Hom(Ky, L), graded by C£(U) = Hom(Ky, L), and with differential the transpose of that of K let C be the associated sheaf, and f : C*(U) -- > C(U) the natural map. It is always surjective; this follows from the fact that K is basic in exactly the same way as the corresponding statement for singular cochains [5, p. 1 6 0 ]. Since K is basic, Ky is a direct summand in Ky(V C U) hence the restriction map Hom(Ky, L) -- > Hom(Ky, L) is surjective; thus C is flabby. Here

Ky

is free, therefore we have a split exact sequence

0 ^ Ext(Hi+1 (K^), L)

-» Kj^HomO^, L)) -» Ham(H1 (Kg), L) -» 0

(see the reference given below 3 *2 (3 )), which Is compatible with the re­ striction maps. In order to obtain the property 1 . 1 (iii), It is enough then to show that f : C !(U) -- > C(U) induces an isomorphism for homology, or In other words that H(ker f ) = 0. This can also.be expressed by saying that the elements of Horn(Kg, L) which vanish on all elements of Ky with sufficiently small supports have zero homology. This assertion Is proved by using the fact that Ky is homotopically fine, In the same way as in the case of singular cochains [3a, Exp. VIII, p. 4]. §5•

Complements

5-1• Let C, C 1 be fundamental sheaves for homology. We have H*(C(U)) = H*(C1(U)) by 1.1 (iii) but since the splitting of the exact sequence is not natural in general, it is not certain that this isomorphism can be so chosen as to commute with restriction maps. It is therefore conceivable that the derived sheaves H(C) and H(C‘) might not be

32

BOREL

CHAPTER II

isomorphic. Nevertheless one can insure uniqueness by taking a narrower class of fundamental sheaves, which will be, however, big enough so that one can define f* in the usual way (see below), namely essentially those which have been constructed in §§3 , k. Let us remark first that the isomorphisms of homological algebra mentioned in §3 are sufficiently "canonical” as to insure that H^(C(U)) does not depend, up to canonical isomorphisms, on the particular injective resolution A of L, which in any case could be normalized by requiring Aq to be the quotient field of L; also in §4, we could as well consider Hom(K, A) instead of Hom(K, L). It is also clear that if K, K 1 fulfill the conditions of §3 , or §4, and if there is a homomorphism f : K -- > K* inducing an isomorphism of H(K^) onto H(K^) for each open subset U, then H(Hom(Ky, A)) and H(Hom(Ky, A)) will be related by an isomorphism commuting with restrictions. Let F be a,fundamental sheaf for cohomology on X. This im­ plies in particular that H^(F) = 0 (i 4 0 ) and that there is an iso­ morphism cp of H°(F) onto the simple sheaf L x X. Let us say that F has local units if given any compact subset Y ( X, there exists a cross section s withcompact support of F° such that for x e Y, s(x) is a cocycle ofthe element of H°(FX ) mapped into 1 by cp. The AlexanderSpanier sheaf, or the sheaves contructed in §4, or the canonical flabby resolution of L in [5] all have local units. If F 1 is another such grating, and if rQdenotes sections with compact supports, then by the fundamental construction of Leray, as modified by Fary [see 1, Exp* II], there are natural embeddings r c ( F ) -----> r c ( F ® F ' ) Hom(Ky, A) where A is an injective resolution of L, and K = rQ(F), F being a fundamental sheaf for cohomology having local units. We have then the PROPOSITION. Let C, _CJ be two restricted fundamental sheaves for homology. Then there are canonical isomorphisms t-jj : H.(C(U)) -- > H ^ C 1(U)) which commute with the re­ striction maps jf(UC V open). Consequently the de­ rived sheaves H(C) and H(C!) are canonically isomorphic. 5.2. The map f* and the exact homology sequence. Let f : X -- > Y be a continuous proper map, F, (respectively G) be a fundamental sheaf with local units on X (respectively Y), and let

CHAPTER II:

§5

HOMOLOGY AND DUALITY

33

f ’(G) be the sheaf induced by G on X. Then P ® f !(G) is a funda­ mental cohomology sheaf on X with local units and there is a natural homomorphism f : r c(G) -- > r c(f* (G) ® F) which defines f* : H*(Y; L) -- > H*(X; L). If X is a closed subset of Y, this map can be identified with the one induced by the natural map rc (0) -- >

rc ^ / rc^-^Y-X

Exp# VI1'

p* 1"^*

We deflne then

f* : H*(X; L) -- > H*(Y; L) as the map induced by the homomorphism Hom(r_G (f1(G) —' ® F), —1 A) -- > Hom(r o —■(G), L), the transpose of f PROPOSITION. Let U be an open subset of X Y = X - U. Then there is an exact sequence

and

f* J?11 (1 ) ... -- > Hj_(Y;L) — > H^XjL) -±— > H^UjL) -- > Hj__1(Y;L) -- > ... Let K be the grating of sections with compact supports of a fundamental sheaf for cohomology on X having local units. Then Ky and K/Ky have the same property with respect to U and X - U. Since A is injective, the exact sequence 0 -- > Ky -- > K -- > K/Ky -- > 0 yields an exact sequence 0 ---- > Horn (K/Ky, A) -----> Hom(K, A)

(2) and (1 ) will be the derived homology sequence of (2 ).

5*3* We finally remark that it is also possible to define homology with coefficients in a sheaf, but we shall not enter into details here, since this would require some modifications of the constructions made in §3 , for instance, in order to get a torsion free fundamental sheaf for homology (a property which the sheaf obtained in §4 clearly has). If this is so, then the condition L - n implies H^.(r(C®B)) = Hn”^(X; H^C)®!^), for any sheaf B by the same proof as for 2 .2 . Also, using injective sheaves, one can dispense with the local units in 5*1* Por all this, see a forthcoming joint paper of J. C. Moore and the author. BIBLIOGRAPHY [1] [2] [3] [4] [5] [6]

A. Borel, "Cohomologie des espaces localement compacts, d'apres J. LerayM. Notes, Ped. School of Technology, Zurich, Switzerland, second ed. 1957* A. Borel, nThe Poincar§ duality in generalized manifolds", Mich. Math. J. 4(1957), pp* 227-239* H. Cartan, Sem. E. N. S. a) 1 9 48 -4 9 , b) 1950-51. H. Cartan - S. Eilenberg, "Homological algebra". Princeton Math. Ser. No. 1 9 , 1956*. R. Godement, Theorie des faisceaux, Act. Sci. Ind. 1252, Paris 1958, Hermann ed. R. L. Wilder, Topology of manifolds, Amer. M. S. Coll. Publ. 32(1949).

Ill:

PERIODIC MAPS VIA SMITH THEORY E. E. Floyd

§ 1.

The Leray spectral sequence

It will be supposed throughout that X is a locally compact Hausdorff space and that G is a finite group acting on X. There is then the space of orbits X/G and the orbit map * : X --- > X/G. The map nis open and proper *“1*x isthe orbit Gx for x e X. The main purpose here is to put the Smith theory of periodic maps [12] in the framework of sheaf theory. Fix a paracompactifying family © (see 1-5*2) which also has Ug€GgM € © wheneverM e © . For example, © may bethe family of compact subsets of X; if X is paracompact, © may be the family of closed sub­ sets of X. There results afamily © for X/G, where N e ©* if _1 * tt N e ©. It may be verified that © is a paracompact ifying family for X/G. To A) and H*(X/G; A) ^ simplify notation, we use H*(X; A) for H*(X; ® — — for H$*(X/G; A), where A is a sheaf on X or X/G. There is the Lerayspectral sequence

for

n

[2, XXI], which

has

as E2-term E|,q = H?(X/G,

L))

and whose E^-term is associated with H*(X; L). Since EP,C1 = o for q 4 0 (since fl^y is finite), then Ep'° = Ep'° = HP (X; L). That is, denoting by A the sheaf UyH ° U " 1y; L), HP (X/G; A) = HP (X; L). We need an explicit form for the isomorphism HP (X/G; A) = HP (X; L). There is it : X --- X/G; there is also a sheaf homomorphism A = UyH°(jt“1y; L) to the constant sheaf X x L = UxeX^°^x^ which is compatible with * in the sense of Cartan [2, XIV 7]. This is the collection cp = (cpx ) of homomorphisms H°(jt”1jtx; L) -- > H°(x; L) in­ duced byinclusion. The isomorphism is induced by the-- pair *, cp. Each

H°(*“1y; L)

contains in a 35

natural way the ring L;

that

36

FLOYD

is, the constant sheaf inclusion X/G x L C A commutativity holds in

CHAPTER III

X/G x L may be considered as embedded in A. The induces a homomorphism HP (X/G; L) -- > HP (X/G; A);

HP (X/G; L)

/ V

l/

Xl

HP(X/G; A) = HP (X; L)

.

We may consider A as given by the presheaf U --- > A(U) = H°(7t“1 U; L) for compact neighborhoods U. Since G operates on jt~1 U, G also operateson A(U), and hence on the sheaf A. On the stalk Ay = H°(tt~1 y; L), g : Ay ------ > Ay is just the homomorphism (g I *_ 1 y) : H°(jr“ 1 y^ L) -- > H°( 1 y; L). We regard G as operating on the right, so that (rg)g’ = 7 (gg' ) for y e A. There are the commutative diagrams X -§-> X Jt V

I

X X L X/G

/\

cp A

cp

I

t*C [Hn (X; L)]°. Since y'e ?'e [Hn (X; L)]° implies jtV'Cy')) = 7r', Image i t* } [H^X; [Hn(Xj L)]®. The corollary follows. «* In a latersection, section, we weuse usea a slight slight modification modification of 2.1, of 2.1, a form a form of the transfer transfer [6]. [6]. Suppose Suppose KK is is aa subgroup subgroup of G. of G. ThenThen K acts K acts on on X and there is the orbit map : X -- > X/K. There is also the unique map *2 : X/K > ---X/G> X/G with with *2 *2 maps an an orbit orbit Kx Kx ofof X/K X/K into the orbit Gx Gxof ofX/G. X/G. 2. it-. PROPOSITION. PROPOSITION.There There is is a homomorphism a homomorphism li1 : HH (X/K; (X/K; L) L) --- >> HH (X/G; (X/G; L) L) such such that n f*2 n f*2 Is multiplication by is by order order G/order G/order K. K. PROOF. From the Leray sequence for the finite-to-one for the finite-to-one map map wp : :X/K X / K-- > X/G, we have H*(X/G, H°(*;1yj L)) L=)H*(X/K; ) = H*(X/K; L). L). NoteNote now now ““1 1 — — 1 1 _ _ 1 1 £~ £~ that Jtp y = y y= =n~n~ y/K. y/K. There is the the sheaf sheaf homomorphism homomorphism © : Uylr(rt21y; L) -- > A = UyH0 (jr“1y; L) L ) defined on each stalk by the the homomorphism H°(jt”1y; L) -- > H ° U “1y; L) induced by ii rr11::iT1y iT1y ----> --- > *21y* *21y* We must compute the image Image of this homomorphism. Since and *’1 *’1 are are finite, the map H°(jt2 1y; L) -- > H°(jr1y; L) may be identified with the cochain map n” : C°(it21y; L) -- > C°(jt“1y; L), where C° denotes the usual 0-dimensional cochain group. It is seen that this is Is an isomorphism into, and has image Image [H°(jt“1y; L)]K . Hence U H°(jt“1y; L) L) ==AK AK and and H*(X/G; AK ) = H*(X/K; L). We define now a 1 : AK -- > A^. For this, it is sufficient to consider any sheaf A upon which G operates (say, on the right). Suppose g-j, ..., gk (k = order G/order K) is a complete set of repre­ sentatives of right cosets of K in G. Define a 1 : AK -- > A by cj1(7) (7 ) - 7g-1 + ... + 7g^« It is Is clear that o' is independent of the choice of g ^ ..., g^. Moreover, ((ccrf7)g r f 7 ) g = 7g-| 7g-|g8 + +

••• ••• + + 7gkg 7gk g

• •

§3

CHAPTER Ills

39

PERIOD MAPS VIA SMITH THEORY

Since g.jg, ..., gkg is also a complete set of representatives of right cosets of K, (o'y)g = a 7 . Hence Image a’ C AG . There are then the diagrams

and the induced diagram if^X/Gj L) = H^X/G; A°)

Hn (X/G; AK ) = Hn (X/K;

L)

yields the stated property. § 3 • An Euler characteristic formula Consider now a homeomorphism g : X -- > X with gp = identity, p prime. Then the cyclic group Zp = {1, g, ..., gp“1} acts on X. Since Zp has no nontrivial subgroups, the isotropy subgroups are either 1 or Zp. Denoting by F the set of fixed points of g, then Zp operates freely on X - F. We first consider the case in which F is empty. 3-1- LEMMA. Suppose that G = Zp, p prime, operates freely on the space X, and that the coefficient ring L is Zp. Suppose that A is the sheaf on X/G of Section 1. If g is agenerator of Zp and t = 1- g is the operator of A, then the sequence A ) tA ) A

)... } tP_1 A 3 0

—1 has A/tA, tA/t2 A, o ..,A D all isomorphic constant sheaf X/G x Zp.

to the

PROOF. The proof of this lemma is obtained by the techniques of Smith and Richardson [11]. Note that = (1 - g)p”1 = 1 + g + ... + gp”1 = a and tP = 0. Now Kernel t = A^, since g generates Zp. Since Z_ operates freely, this is also seen to be Image a : A -- > A. P -q_1 i Hence Kernel t = Image . Suppose it has been shown that Kernel x Image and consider the proposition Kernel = Image . Since = 0 , Image C Kernel t1+1 . Suppose T^+1(r) = 0. Then T^(xr) = 0 and tp = . Hence x(r ) = 0 and r r ’ = TP_1rn. Hence r = (r1 + T^r"). Hence for 1 < i < p - 1 , Kernel = Image in particular, Kernel tp_1 = Image

CHAPTER III

FLOYD

ko

Consider now the homomorphism tP”1-1 of t^A onto tp ~1A. T^r is In the kernel, then TP_i”1(Tir) = TP-1r = 0 so that r = xr1. Hence the kernel is t^+1A and T^A/Ti+1A « tp”1A « A^ ~ X/G x Zp.

If

A slightly different proof of the following may be found in [5]. 3*2. THEOREM. Suppose the cyclic group G = Zp, p prime, operates freely on X, where dim^X < » and H*(X; Zp) is finite dimensional. Then H*(X/G; Zp) is finite dimensional and x(X; Zp) = p • x(X/G; Zp ) where x(Y;Z^) denotes z(-1 )ndim H^X; Z^). PROOF.

Let A1 = A =

t^A,

o < i £ p - 1.

There Is the filtration

A0 5 A1 5 ... 3 A15" 1 ^0

of the sheaf A. Given such a finite filtration of the sheaf A, there is a spectral sequence {E^^} with_ h s +^(X/G; A3/As+1) and with E^ associated with afiltration of H*(X/G; A) = H*(X; Zp). By 3 .1 , E3>t _ Hs+t(X/G; Zp) for 0 < s < p - 1 . Of course, E^>t = 0 for s < 0 or s ^ p . If H (X/G; Zp) Is finite dimensional then we have

x(E1 ) - x (E2 ) = . . . = X(EJ = x(X; Zp) and x(E1 ) = s ( -l )3+tdim E3 ,t = p • x(X/G; 2^). Now

jt

Hence it is sufficient to prove H*(X/G;Zp) : X ---> X/G Is a local homeomorphism. Since

dimension is a local property (see 1 -2 .3 ),

finitedimensional. dimz X < » and since P dim^ X/G < ®. Suppose now that

it has been shown that dim HJ (X/G; Zp) < °° for all dim E^'^ < * for s + t > n and dim E^'^ < 00 for E ^ is Kernel d£'n ; E°,n --> E ^ n’r+1 . Since

j > n. Then s + t > n. Now E ^ n“p“1 is finite

dimensional, then dim E°'^ < » if dim E°'n < 00. Since dim E ^ n < 00, it follows that dim E°'n = dim Hn (X/G; Zp) < 00• The induction follows. It should be noted that the finite dimensionality of follows more generally from the Cartan-Leray spectral sequence been included here for completeness. §4.

Hn (X/G; Zp) [1 ]; it has

The Smith sequences

Here we confine ourselves to the action of a cyclic group G = Zp, p prime, on a locally compact Hausdorffspace X. Denote by A the sheaf of Section 1 , by g a generatorof Z^, and by t , a the endomorphisms

CHAPTER III:

41

PERIODIC MAPS VIA SMITH THEORY

1 - g> 1 + g + ••• + gP”1 °f A. If one of these is denoted by p, the other will be denoted by p. The coefficient ring L is assumed to be Zp. Denote by F the set of stationary points of Z^; F is also the set of fixed points of g. If f : A -- > X/G is the natural map of the sheaf, then Ay (U open in X/G) denotes f“1U U {Oy : y € X/G) and A^ (M closed) denotes A/Ax_^. We develop a cohomology form of the special homology groups of P. A. Smith. These were originally defined by Smith (for references, see [12]) and used by Smith, Richardson [10], Liao [9], and the author [5] to develop a theory of periodic mapso We confine ourselves here to a brief global treatment, as the topics will be dealt with later using the spectral sequences of Cartan-Leray and Borel [1]. 4.1. LEMMA.

The sequence P ©

T)

o -- > PA -- > A ----- > pA © Ap -- > 0

is exact, where A -- > A is Inclusion and where t) : A -- > A-p is the quotient homomorphism. PROOF. It is sufficient to verify exactness on each stalk. y e X/G - F, this reduces to exactness of

For

0 -- > pAy -- > Ay -- > PAy -- > 0 which pAy =

has been shown in the proof of 3 .1 . For y e F, it is seenthat pAy = 0 , so thatexactness reduces to exactness of 0 --- >

0 --- > Ay -> Ay -> 0

k.2. DEFINITION. The groups Hn (X/G; pA) will be called the Smith special cohomology groups. When no confusion results, denote them simply by Hn (p). The exact cohomology sequence ...-> H^X/G; pA) -» if1(X/G; A) -» H^X/G; p A ) © H n (X/G; Ap ) ^ ... of (4.1) becomes .. . -» H^p) -> H^X; Z^) -» Hn (p) © H n (F; Zp) -» Hn+1 (p) -> ... and will be called a Smith sequence. Note that

oA

is the constant sheaf

(X/G x Zp)x/G-F

so that

.

FLOYD

CHAPTER III

Hn(0) = H£*(X/G - F; Z^). Theorems of the following type have been proved in [5], and by Heller [8]. f 4.3. THEOREM. If G = Z p where dim^ < °°, then P dim if1(X/G - F; Zp) + for any integer PROOF.

prime, acts on X,

dim H^F; Zp)
H^cr) + Hn (F; Zp ) -- > H ^

t)

(n + 1 )

Hn+1 (X; Zp) ---> Hn+1 (t ) + Hn+1 (F; Zp) -- > ^ ( a )

(n + 2)

r f ^ X j Zp) ---> Hn+2(a) + Hn+® (F; Zp ) -- > Hn+3(T)

(n + 2i) Hn+2i(X; Zp) -- > Hn+2i(a) + Hn+2l(F; Zp) -- > Hn+2i+1(x) Since

A ---> B > C

exact implies

.

dim B ^ dim A + dim C,

we obtain [n]

dim ^(a) +dim H^Fj Zp )

[n + 1 ]

dim

dim E p ^ a )

Hn+1 (t )+

£ dim H^X; Z^) + dim Hn+1(i:) dim Hn+1 (Fj

Zp) £

+ dim Htl+2 (F; Zp ) < dim r f ^ X ;

dim Hn+1 )X;Zp )+ dim

Zp) + dim Hn+3(x)

[n + 2i] dim Hn+2l(a)+ dim rf1+2i(F; Zp) < dim Hn+2i(X; Zp) + dim Hn +2l +1(t If dim Hn+2l+1(T) < co, it is seen by going backwards step by step that all terms are finite. If n + 2i > dim^ X, then all terms of [n + 21] are

0.

The terms

Hn+2i(p)

are

0

since

dim (X- F) and since pA is essentially asheaf on clusion follows by addition and cancellation.

dim^(x/G - F)

H£+1 (X - F; ^ ) -- > H£+1 (X; Zp)

we see that Hn+1(X - F; Zp) is finite dimensional for all 3.2, X$(X - F; Zp) is divisible by p.

n.

Hence, by

Now from the cohomology sequence (l ), X$ (X;

Z p ) - X$ ( F ; Z p ) + X$ (X -

= X$(F; Zp) mod p

Fj

Zp )

.

We obtain now the well-known Smith theorems [12], as well as a dimensional parity result [5]. 4 .5 . COROLLARY. Suppose that Zp, p prime, acts on X, where dimz X < » and where H*(X; Zp) = H*(Sn; Zp ). P There exists an - 1 < r < n such that H$ (F; Zp) = H (Sr; Zp), where the (-1)-sphere is interpreted as the empty set. If p Is odd, then n - r is even.

4.6. COROLLARY. Suppoze that Zp, p prime, acts on X, where dimz X < °o and where H^(X; Z^) = 0 for 0, H°(X;Zp) = Zp. Then H^F; Zp) = 0 for i > 0,

F

Both of the corollaries follow §5«

Is not empty and H°(F; Zp) « Zp . easily from (4,3) and(4.4).

Orbit spaces of finite groups

5.1 PROPOSITION. If the finite group G operates on the locally compact Hausdorff space X, then dim^X/G = dim^X. PROOF. Note first that if A Is a closed subset then dim-j-X = max (dim-j-A, dimL (X - A)}. For if U is open the exact sequence ...

— > h£ ( u - a nu ) — > h £ ( u ) — > h £ ( a n u ) — > . . .

where coefficients are in L. order

ofa space X, in X wehave

Hence the equality holds.

Let n = order G, and let A^ denote the set of all Gx ^ i, where Gx denotes the isotropy subgroup. Then

x €X

with

CHAPTER III

FLOYD X = A,

3 A 2 D ... ) A n ) A n + 1 = »

is a sequence of closed subsets of GgX = gGxg

—

1

.

Moreover,

all subgroups

H

of order

i

: X ---> X/G

A^ - Ai+1

each

and where

are pairwise disjoint and open in jt

X;

- Ai+1 = U^B^

A.

B^- = {x : Gx = H).

A^ -

jtA^ - ^rAi+1 .

B jj

B a n d

hence of

Hence, since dimension is a local property,

dim^(A^ - A^+1 ) = dim^(jcA^ - *A^+1 ).

Moreover

An

dim^A^

jrAn .

The

also the orbit map

is a local homeomorphism on each

onto

onto

is G-invariant, since

where the summation is over

Inductively, suppose

it

is a homeomorphism of

= dim^rrA^ .

Then

dim^jrA^ = max {dim^itA^^, dim^JtA^ - *A^+ 1 )} = max {dimLAi+1, diro^A^ - A1+1 )}

= dim^A^ i = 1

For

we receive

theorems for compact Lie groups

dirn^X = dim^X/G. G

References to similar

of transformations may be found in

Yang [13]. 5.2. on

THEOREM. X,

where

Suppose that a finite group dim-^X < o 00o and

H^(X; L)

generated (respectively trivial) for all L

Is either the ring of integers is

of characteristic

p.

Then

Z

i I ^^ N, N,

PROOF. L =

they follow from 4.3*

I i ^ N.

then the assertions follow from 2.3* 2.3Consider now the case

From the sequence

where where

is finitely

We prove the theorem first in the case

(q ^ p),

acts

or a field

H^(X/G; L) H^X/G;

generated (respectively trivial) for all

If

G

is finitely finitely

G = Z^ If

(p

prime).

L = K^,

L = Z.

0 ---> Z -E-> Z --->

---> o, p

prime, we ----- > 0 , p

obtain ... ---> Hn (X; Z) ---> Hn (X; 2^) ---> Hn+1 (X; Z) ---> ... Hence H^X ; and

Hn (X; Z)

and

Hn+1(X; Z)

finitely generated [trivial] implies

Zp) isisfinitely finitelygenerated generated [trivial]. [trivial]. By Byk .k3. >3 >

Hn Hn (X/G (X/G -- F; F; Zp Zp ))

Hn (F; Z^) Z^) are arefinitely finitelygenerated generated [trivial], [trivial], and and hence hence so so is

Hn (X/G; Z p L ).

.

Consider the exact sequence

is

prime, w

§5

CHAPTER III:

PERIODIC MAPS VIA SMITH THEORY

... -- > Hn (X/G; Z) -2-> Hn (X/G; Z) -2_> Hn (X/G; Z ) A

Hn (X; Z) of cohomology o -- > Z -- > Z -- > Z^ -- > 0, to which the maps of 2.1 have been added. Since Tp'iX; Z) is finitely generated [trivial], so is Image p. Since H^X/G; Z^) is finitely generated [trivial], so is Image q. Since Kernel q and Image q are finitely generated [trivial], so is Hn (X/G; Z). Hence the theorem is true for G = Z , p prime. Suppose now that G is finite solvable of non-prime order r, and that the theorem has been proved for all solvable groups of order < r. Then G contains a non-trivial normal subgroup K, which is solvable since Gis. Moreover, G/K is solvable, and K and G/K are of order < r. Now K acts on X and there is the orbit map jt1 : X -> X/K. point Kxof X/K is acted.upon by G, since g(Kx) = Kgx e X/K. Under this action of G on X/K, K acts trivially on X/K. Hence G/K acts on X/K. It may be seen that we can identify X/G with (X/K)/(G/K). Let *2 : X / K -- > X/G be the natural map. By induction hypothesis, H^X/K; L) is finitely generated [trivial] for i ^ N as then also is H^(X/G; L). Hence the theorem follows for solvable finite groups G.

A

Suppose now that G is an arbitrary finite group, with order ni nk G = P1 ••• Pk • There are then Sylow subgroups K 1, ..., K^ of G with order K • = p.*5 *. Since K. is solvable, Hi (X/K-; Z) is finitely J u J J generated [trivial] for i ^ N. We now use a device due to C. N* Lee (unpublished). There are the maps X whose composition is

*•

•> X/K

X/G

Hence there is

na = *2,1 + ••• + n2,k : H*(X/q5

----- > H*(X/K1; L) + ..

Note for future reference that *a is canonical. That is, if f : X -- > X f where X 1 is locally compact, where G operates on X 1 and f is equivariant, and where f is appropriately related to the support families for X and X ’, then commutativity holds in H*(X' /G; L)

\l

2H*(X'/K1; L)

I'

H*(X/G; L) — -- > SH#(X/Ki; L)

FLOYD

C H A P T E R III

w h e r e the ve r t i c a l h o m o m o r p h i s m s are i n d u c e d b y W e show n o w that *

ke r n e l t h e n

jt2 ^(y) = o

*

is injective.

(all

/

i).

i.

n 1

It f o l lows that

1

7 = 0.

^ -4

Hence

a r b i t r a r y c o e f f i c i e n t group.

Si n c e f o r

f i n i t e l y g e n e r a t e d [trivial],

so is

5 *3 o

COROLLARY.

group and

G

H (X; Z)

x(X) them. PROOF.

5 .2 .

That

ye

H*(X/G; L)

is

in

i

, -I

... P k

it ^ N,

n>

\

Jr = 0

is i n j e c t i v e f o r a n each

1

H ' ( X / K y L)

is

H*~(X/G; L).

Su p p o s e that a f i n i t e sol v a b l e

operates f r e e l y o n

H*(X/G; Z)

If

Hence by 2 . 3

(order G / o r d e r 1 ^ ) 7 = ( P 1 • •• P i _^’P l+1 f o r all

f : X --- > X ’.

X,

where

is f i n i t e l y gen e r a t e d .

d i m ^ X < 00 Then

is f i n i t e l y generated, and

x(X/G),

are i n d e p e n d e n t of the fi e l d u s e d to d e f i n e Moreover That

x(X), x(X/G)

x(X) = (order G)x(X / G ) .

H*(X/G; Z)

u n i v e r s a l c o e f f i c i e n t theorem. b e e n p r o v e d i n 3 *2 .

is f i n i t e l y g e n e r a t e d f o l l o w s f r o m

is i n d e p e n d e n t of the f i e l d f o l l o w s f r o m the For

G = Z^, p prime,

the t h e o r e m has

It fo l l o w s i n th e g e n e r a l case b y i n d u c t i o n just

as

i n 5 .2 . BIBLIOGRAPHY [1]

A. A. Borel, Borel, N o u v eNloluev edlelmeo ndsetmroantsitorna tdiTu o nn dt Tu h eno rtehmeeo rde e m eP-deA.P.Smith, A. Smith, Comment. Math. Helv. 29(1 955 )> 27-39*

[2]

H. H. Cartan, Sem. E. N- S. 1950 - 5 1 .

[3]

P« P« E.E.Conner, Conner,C oCnocnecrenrinnign gthe thea catcitoino nofofa af ifniintiet egroup. group.Proc. Proc.Nat» Nat» Acad. Sci. 1*2 ( 1 9 5 6 ), 3^9-351 •

[k]

B. Eckmann, Cove s anBde tBteit tnumbe i numbe Bull. Amer. Math. Soc. B. Eckmann, Cove r i nrgisn gand r s .r s .Bull. Amer. Math. Soc. 55(19^9), 95-101. 55 19 9 9510 1.

[5]

E. E. E. Floyd, E. Floyd, On p e rOn i o dpiecr imoadpisc and m a p sthe andE uthe l e r Ecuhlaerra ccthearriascttiecr iof s t i c of a s s o c i a t e d spaces, Trans. Amer. Math. Soc. 7 2 ( 1 9 5 2 ), 1 38 - 1 ^ 7 .

[6]

E. E. E. E. Floyd, Floyd, Orbit Orbit spaces spaces of of ffiinniittee ttrraannssffoorrmmaattiioonn groups, groups, I, I, DDuukkee Math. J. 20(1953), 563 - 56 8 .

[7]

A. A. GGrroothendieck, thendieck, Sur Sur qu queellqquueess ppooiinnttss dd'!aallggeebbrree hom homoollooggiiqquuee,, TToohhookkuu Math. J. 9(1957), 119-221.

[8]

A. A. Heller, Heller, H o m oHloomgoilcoagli craels orleustoilountsi oof n s cof o m pcloemxpelse xweist hw ioperators, t h operators, Ann. of Math. 60(195*0, 283 - 3 0 3 .

[9]

S. S. D. Liao, D. Liao, A t h e oAr etmh eon o r epme roino dpiecr itordaincs ftorramnastfioornmsa tof i o nhso mof o l ohgoym o l o g y spheres, Ann. of Math. 56(1952), 68- 8 3 .

Cartan, Sem. E. N- S. 1950 - 5 1 .

( ^ ),

§5

CHAPTER Ills

PERIODIC MAPS VIA SMITH THEORY

[1 0 ] M. Richardson, Homology characters of symmetric products, Duke Math. J. 1(1935), 50 -6 9 . [11]

M. Richardson and P. A. Smith, Periodic transformations on complexes, Ann. of Math. 3 9 (1 9 3 8 ), 611-633*

[12]

P. A. Smith, Fixed points of periodic transformations, Appendix B in Lefschetz, Algebraic topology, 1942.

[1 3 ] C. T. Yang, Transformation groups on a homological manifold, Trans. Amer. Math. Soc. 8 7 (1 9 5 8 ), 2 6 1 -2 8 3 .

47

IV:

THE ACTION OP

OR T1: GLOBAL THEOREMS A. Borel

§ 1.

Transformation groups

We collect here for future reference some elementary facts and notation. Let G be a topological group. A G-space is a space on which G operates (continuously of course). Then Gx = {g € G, g • x = x) de­ notes the stability or isotropy group of x e X, and for a subset A of X, G • A = (g • a, g e G, a e A) is the union of the orbits G • a of the points of A. It is open if A is. The space of orbits is denoted by X/G, or when G is fixed once and for all, by X 1, and : X -- > X ! is the natural map assigning to x e X its orbit. X 1 is assumed to be endowed with the quotient topology, and is open and continuous. The fixed point set of a subgroup H in an H-invariant sub­ space will be denoted F(H; A). Two orbits are said to be of the same type if their stability groups are conjugate (by inner automorphisms) in G. We say that the orbit structure is finite (respectively locally finite) if the stability groups belong to a finite number of conjugacy classes (respectively if each point has an invariant neighborhood with finite orbit structure). When G is compact, the following facts are easily proved: X/G is Hausdorff (respectively locally compact) if X is. The natural map G/Gx -- > G • x isan isomorphism. If A is closed in X, G • A is closed, hence ^ isclosed, and therefore also proper (since the orbits are compact). 1.2. Let N be a closed invariant subgroup of G. Then X/N is in an obvious way a (G/N)-space G/N and p : X -- > X/N be the natural maps. Then, clearly, a(Gx ) = (G/N)p(x ) for any x e X. In particular: if N contains all isotropy groups, G/N acts freely on X/N; if G is connected and Nis finite, then p is a homeomorphism of F(G; X) onto F(G/N; X/N). 49

50

CHAPTER IV

BOREL

1 .3 . Assume that G also operates on a space Y. We denote by XXqY the orbit space of X x Y under the "diagonal action” g(x, y) = (g • x, g • y). In other words, it is the quotient of X x Y by the equivalence relation (x, y) « (g • x, g • y). The canonical pro­ jections of X x Y onto its two factors induce projections 7^, it of XXqY onto X/G and Y/G respectively. Let y e Y and y 1 be Its orbit. Then it is easily seen that (y1) may be identified with X/G • Moreover, if Y is a locally trivial principal G-bundle, then jt2 is the projection of a locally trivial fibre bundle, with typical fibre X and structural group G. If G acts trivially on X, then XXqY = XX(Y/G) and j^, *2 are the natural projections• There are of course analogous statements with X and Y Interchanged. Here we have assume G to operate on the left on X, Y. If we are given a space E on which G operates on the right, we consider instead the left operations defined by g • x = x • g"1(x e E, g e G ). * Let G be another topological group, \ : G -- > G a homo•if * morphism, and X be a G -space. A X-equivariant map f : X -> Xis a continuous map such that f(g • x)= \(g) • f (x). It induces a map f : X/G ---> X*/G*. IfG = G* and \ is the identity, then fis said to be equivariant. Let Y be another G -space and h : Y --- > Y •H* -Xbe a x-equivariant map. Then f x h : X x Y -- > X x Y is a X-equivariant; it induces a map n : Xx^Y -- > X x q *Y and we have a commutative diagram X/G Y/G

^ | X*/G* < - A - X*XQV

^g Y*/G*

§2• Some remarks on the cohomology of A module S graded by submodules type if each S^ is finitely generated.

S^

. B^

is saidto be of finite

2 .1 . Cohomology of finite Let G be afinite group, - - - - - - C*groups. -— A a module on which it operates. A denotes the set of elements fixed under G, and H (G; A) is the cohomology group of G with coefficients in A (see, e.g., [4] Chap. XII for this and the results mentioned below). (1 ) H°(G; A) = AG; for (n = ord. G).

be gm

i > 1 , we have

nH^G; A) = o

(2 ) Let G = Z^, generated by an elementg. Let NA, ^A,IA respectively the subgroup generatedby the normsg * a + g 2 * a + . o . + •a, by the elements of norm zero, and by the elements g • a - a.

§2

CHAPTER IV:

THE ACTION OP Zp OR T1: GLOBAL THEOREMS

51

Then H^G; A) = AG/NA (i even, ^ o); H^(G; A) = ^A/IA (3) Let p prime, m = ps, and A = Z^. acts trivially on A. We have H*(Z2; Z2 ) = Z2 [a] H * ^ ; Zp)

= A [a] ®Zp[b]

(d°a =

(i

odd)

•

Then, clearly,

G

1)

(d°a = 1, d°b = 2; m = ps, m + 2 )

Thus, if m 4 2> the multiplication by and annihilates

a is injective on

.

H2l(Zm; Zp )

v also recall that if G = T , the k-dimensional torus, H*(Bq ; L)is a ring of polynomials in kvariables of degree two. 2 .2 . We

2.3* If G- is a compact Lie group, H (B^; L) is of finite type. For G connected this follows from the fact that in the universal fibering, the cohomology of the total space and the fibre are of finite */ % type and the sheaf of the cohomology groups H (G; L) is constant. In the general case we use the fact that B (G° identity component of G) G is a regular covering of B^ with group G/G° and that the cohomology of a finite group with respect to a finitely generated module is of finite type, since it can be computed by means of a chain complex of finite type. 2.4. PROPOSITION.

then

Let

L = Kp, A,

and

G 4 (E).

(a) If G is finite, and L = Kp with p not dividing H*(B^; L) is trivial (i.e., H^(B^; L) = 0 for I ^ 1).

ord G,

(b) In all other cases, there are arbitrary large i such that H ^ B qJ L) 4 0, and contains a non-zero free L-module when G Is infinite. PROOF (patterned after an unpublished remark of Serre about finite groups). Let M be a compact connected orientable manifold on which G acts freely, and on the cohomology of which G acts trivially. This always exists, take for instance a group S0(n) In which G is embedded by means of a faithful representation. Let n = dim M. Since M is a principal bundle, there is a spectral sequence leading from E2 = H(BQ; H*(M; L)) to H*(M/G; L), as will be recalled in 3.4. * Assume first H (Br; L) to be trivial. It follows from the ■X* *)(■ spectral sequence that * : H (M/G; L) -- > H (M; L) is an isomorphism, hence dim M/G = dim M, and G is finite; moreover, since the Brouwer degree of it is then equal to ord G, n can be an Isomorphism only if

CHAPTER IV

BOREL

52

L = Kp with p not dividing ord G. Conversely it is known that in this case H (B^; L) is trivial (2 .1 ). This proves (a) and shows that in case (b), there exists a j ^ 1 such that H^ (B^; L) =( 0 . In the spectral sequence mentioned above, we have Ep'n = H^B^; L) since Hn (M; L) = L and is acted upon trivially by G. Therefore if the i's for which H^B^; L) 4 0 were bounded from above, the maximum cocycle argument would show the existence of some m > n for which if11(M/G; L) 4 0 in contra­ diction with the fact that M/G is a compact manifold of dimension < n. There are therefore arbitrarily large i for which H^B^; L) 4 °; this ends the proof of (b) if L is a field, or if L = Z and G is finite. If now G is infinite, and L = Z, the last part of (b) follows from the fact that H1^ ; Z) is finitely generated and from the firstpart of (b) applied to the case L = KQ. §3•

The space

X^

3.1. Let G be a topological group. E^ and B^ denote respectively a universal G-bundle and a classifying space for G. Thus Eq is a principal G-bundle with trivial cohomology in all dimensions, and Bq is its base space (cf. for instance [1], [9])‘ If G is discrete, thenH (B^, A) is the cohomology group of G in the sense mentioned in 2.1 and will accordingly be denoted by H (G; A). DEFINITIONc Let X be a space on which denote by X^ the space Xx^E^ (see 1 .3 ) and by maps of X^ on X/G and B^ respectively.

G

operates. Then we the natural

From the remarks made in 1 .3 , it follows that *2 is a fibre map, in a fibration with typical fibre X and structural group G, and that if x € X, x 1 = jt(x), then jt“1(x') = E^/Gx, therefore ^ ( x 1) may be identified with a classifying space for Gx . In this hook we are interested only in the case in which G is a compact Lie group and X is locally compact, and in order to avoid some technical and irrelevant complications we shall assume that E^ is compact but then N-universal (i.e., H^Eq) = 0 for i < i < N) for some integer N. It is then understood that in all statements below, i > dim X or i > n should, strictly speaking, be replaced by dim X < i < N or n < i < No Since N canbe chosen to be arbitrarilylarge, this convention will not lead to any confusion. For finite G, it could be avoided al­ together by taking as E^ a G-free and acyclic complex, finitely generated in all dimensions, and by replacing X by some fundamental grating on which G operates. 3 .2 . Let

G*

be a compact Lie group,

X

: G -- > G*

a continuous

§3

CHAPTER IV:

THE ACTION OP Z

OR T 1 : GLOBAL THEOREMS

53

homomorphism, Y a G -space, and f : X ------- > Y a \-equivariant map.To X there is associated an equivariant map p of Eq into Eq * for suitable Eq , Eq *. This follows directly from Milnor’s construction using joins, and can also be seen as follows: Let Eq ,Eq * be universal bundles. Then Eq x Eq * considered as G-space under the action g(u, v) = (g • u, x(g) • v)) is also a principal G-bundle and is clearly acyclic, hence may be viewed as a universal G-bundle. We then define ia as the projection on the second factor. It induces a map p(x) : Bq — > Bq *. Then f x p : X x Eq -- > Y x E ^ is X-equivariant, hence defines a map f ^ : Xq -- > Yq * and we have clearly a commutative diagram x/G

Bq p (x )|

f,

\>

V

Bq.* X/G*

.

If G is an invariant subgroup of G , X the inclusion map, X = Y and f the identity, then f and p(x) are the projection maps of principal (G*/G)-bundles. 3«3* Xq is a bundle over Bq , with fibre X, structural group G, projection map Jt . Assume that there Is at least one fixed point a. Since it is invariant under the structural group, the fibre over y e Bq has a well defined point a^. corresponding to a under any of the allow­ able homeomorphisms of jt“1 (y) over X, and y --> a^Is then clearly a continuous cross section. Consequently: if there is afixed point, then *2 : H (Bq ; L) -- > H (Xq ; L) is injective and Im *2 is a direct summand for any coefficient ring L. If a is viewed as a point of X/G, then the above cross section may be identified with(a), therefore the natural restriction * *#■ _ 1 map H (X^; L) -- > H (*“ (a); L) is surjective. It follows that * * * Im r x f contains the image of Im r ^ (x) H (Bq ; L) under the canonical map of

H*(F; L ) ® H * ( B q ; 3»^.

l

) into

H*(F x Bq ,* L).

H^Bp ; L) = o for 1 < I < N x . . and all x e X. Then by the Yietoris mapping theorem H (Xq ; L) = H (X/G; for 1 < i < N; the same is true in cohomology with compact supports if X is locally compact. There is then a spectral sequence (that of ) leading to the cohomology of X/G (up to N), and in which E2 = H*(Bq ; H * (X; L)) (or E2 = H*(Bq ; H*(X; L)) as the case may be). We mention three cases to which this remark applies: (a)

Applications. Assume that

X Is a principal G-bundle.

See [1] for numerous

L)

applications.

5^

BOREL

CHAPTER IV

If G is finite, we get in this way the Leray-Cartan spectral sequence for regular coverings. (b) G is a circle, without fixed points in X, field of characteristic zero. In that case Gx is finite, hence H*(Bg ; L) is trivial by 2.1.

and

L

is a

-A.

(c) G is finite, L is a field of characteristic zero or prime to ord G.In this case, moreover, 2.1 implies that in the above spectral sequence ofwe have= ofor p ^ o, and E°,q- = H^(X; L)G . Hence we have H*(X/G; L) = H*(X; L)G , as was already proved in III, 2 .3; it would also be easy tocheck here that the iso-Xmorphism is induced by jt^. 3«5« Let Y be another space acted upon by G and f : X — > Y a continuous equivariant map. Then (3.2), f induces a map f^ : X^ — > Y^ compatible with the projections jc2 and inducing the identity on B^; it will therefore also yield homomorphisms of the corresponding spectral sequences. This applies in particular if Y is a G-invariant subspace of X, hence Y^ may then be Identified with a subspace of Xq. If Y = F is the set of fixed points of G in X, then we have a natural inclusion of Fq = F x Bq into Xq and the restrictions of jc2 to F^ are the canonical projections on the two factors. 3.6. PROPOSITION. Let G be a compact Lie group acting on the locally compact space X, and let F be a closed invariant subspace. Assume that for x L) Is an Isomorphism for n < I < M. Let

U = X - F

and

U ! = X 1 - F 1 be its image in

X 1. Then we

can apply 3.^ to *2 : UG -- > implies that H ^ ( U L ) = L) for 1 < i < M, hence H^(Uq.; L) = 0 for n < I < M. Clearly Uq = Xq - Fq and the proposition now follows by use of the cohomology se­ h J(U« ;

quence with compact carriers for

Xq mod Fq.

3.7. Applications. The assumptions 3.6 are fulfilled in 3.7* Applications« The assumptions of 3*6 areoffulfilled in each of the following cases: (a) dim^X = n, G acts freely outside F (because dim^X1 - F ! < din^X - F. This follows here simply from the fact that each point y e X* - F 1 has a closed neighborhood homeomorphic to a closed subset of X. For a more general result see [10] or (VIII, 3»16).) (b) G = L = Zp (p prime), dirripX = n, and F the fixed point set; because then G acts freely outside F, and we are in case \

CHAPTER IV:

dimj-X = n,

(c) and

P

THE ACTION OF Zp OR T1: GLOBAL THEOREMS

55

P is a circle, L a field of characteristic zero, the fixed point set. The assumption on H (Bq. ; L)

follows from 3.4 (b).

The other one from the fact that

dim^X' < dim-j-X

[1 0 ].

3 .8 .

Proposition 3 . 6 is also valid for a not necessarily locally compact space in ©-cohomology, where © is a paracompactifying family containing the saturation of its elements. The proof uses -the Vietoris mapping theorem for ©-cohomology and proper maps, and the co­ homology sequence in ©-cohomology, but is otherwise the same. The assumption on H (X1 - P) must then be made for ijr-cohomology where y Is the image of © (which is paracompactifying under the assumption made on ©). 3*9* REMARK. All our discussion will center around the space and the remarks 3 .6 , 3 . 7 will be basic. Similar arguments have been used by Conner [5] when G is a circle, in rational cohomology. For an algebraic analogue when G is discrete, see [7, Chap. V]. The space Xq and the embedding F(X; G) x Bq C Xq were also mentioned to the author by A. Shapiro. The proof of Smith's theorem 4.3 is also related to that of [2].

Xq ,

§4.

The fixed point set of a prime period map in a cohomology sphere

The present section Is concerned with results already proved in III, except for 4.5, 4.7, and Is meant mainly as an illustration of the method underlying this lecture. In this paragraph, p is a fixed prime. Unless otherwise stated, cohomology is taken with respect to Z and * -xP coefficients are omitted. We also write H (G; A) for H (Bq ; A) when G is finite. Except in 4.5, 4.6, G is a cyclic group of prime order p acting effectively on a space X, F its fixed point set and (Ep ) the spectral sequence of the fibering *2 : X q ---> B q . 4.1. Let dim^X = n. (I ^ n + 1 ). H^(Xq ) =

(The first

H^(F x Bq ) for

H ^ F x Bq ) = H*(F)

for

Then

dim H*(X) ^ dim H*(F) = dim H1 ^ )

inequality Is contained in III, 4.3). By 3*7(b) I ^ n +1. But, by 2.1 and the Kunneth rule, I ^ n, whence the equality of 4.1. On the other

hand (1 ) dim H^(Xq) = Sgdim E^-3,s £ z dim e|-3'3 = z dim Hi-S(G; H^X))

and the inequality follows from 2.1. The proof also shows that dim H*(X) = dim H*(F) if and only If E2 = E^ and G acts trivially on

56

CHAPTER IV

BOREL

H*(X). If G acts trivially on H*(X), then dim H*(X) = dim H*(F) if and only if E2 = Em. The if part follows from the preceding proof and 2.1. Conversely, if E2 4 E^, then, using 2.1 (3 ), it is easily seen that there are elements of arbitrary high degrees on which some differ­ ential is non-zero; therefore dim E^”s's < dim E2“s's for suitable i > n and s \ 0, and we have a strict inequality in (1 ). 4.2. If X is acyclic over Z^, then in (Ep) we have only the fibre degree zero; therefore H^(Xq )= E ^ ° = E2'° = H^(G; Z^) = Z^, hence dim H (F) = 1, and F is non-empty and acyclic (III, 4.6). 4 .3 . Assume now that X is a homology n-sphere over Z^. We want to show that F is a homology r-sphere (- 1 < r ^ n) over Z^, where (-1 )-sphere stands for the empty set.

First let n = 0 . Then components, X1, Xg. If these p = 2 of course), then F is homology o-sphere.

X

From now on assume n ^ 1. We distinguish three cases.

consists of two acyclic connected are permuted (which is possibleonly if empty. If not, 4.2 shows thatF is a Then, by 2.1 (3 ),

E2 = H*(Z^) ®H*(X).

(i) If E2 = E^, then (4.1 ) dim H#(F) = dim H*(X), a non-empty homology sphere.

thus

F

is

(ii) If n is even and p odd, then E2 = E^. This is the wellknown argument for sphere fiberings: Let z be a generator of iPCX). Since H*(X) = H*(Sn ), we have E2 = En+1 and En+2 = E^, and E2 = E^ if and only if dn+1 (1 ® z ) = 0 . (This is of course for arbitrary p, n. )If now nis even and p odd, then dn+10 ® z2 ) = 2 dn+1z(x)z = 0 since z2 = 0 hence dn+1(l (x)z) = 0 . (iii) If E2 4E^, then F is empty. In fact, since H*(X) is generated by transgressive elements, our assumption implies that *2 : H (Bq ) -- > H (Xq ) is not injective and our contention follows from 3.3. This shows that F is a homology r-sphere. That r < n was proved in III. Aproof within the present frameworkmay be found in ([2], 3 .6 ) and will not be repeated here. 4.4. Under the assumptions of 4.3, if p is odd, then n - r is even (III, 4.5)- If n = 0 , see the beginningof the proof of 4.3• If F is empty (i.e., If r = - i ) , then n must be odd by (i), (ii) * above. Now let F be non-empty and n > 1. The rings H (X^) and * -#■ '■* H (Fq ) are modules over H (Z^) and these operations commute with the

§4

CHAPTER IV:

THE ACTION OP ^

OR T 1 :GLOBAL THEOREMS

57

restriction map H*(Xq ) --- > H*(Fq ) (XVI, §1 ). Leta be the operation defined by the generator of degree 1 of H (Z^) (see 2.1 ), on these rings or on (E ). Let i be even. If r is odd (respectively even), then a annihilates ) (g)Hr (F), (respectively is injective on H^(Pq ). Since F is assumed to be non-empty, we have (4.3 (iii)) E0 = E , hence on E we see similarly that if n is odd, (respectively 00 n "n m even) then a annihilates E^” , (respectively is injective on the terms of total degree n). This implies easily that a : H^(Xq) -- > H^+1(Xq) is not injective (respectively is injective) for n odd (respectively even). The fact that n and r have the same parity follows now because •#* the restriction H (Xq) > H (Fq)commutes with a and is an iso­ morphism in degrees i ^ n-+ 1. * -* 4.5* By a trivial induction, the inequality dim H (X) ^ dim H (F) and 4.2, 4.3, 4.4 remain true if G is a finite p-group. Also, (in view of 2.1 ), 4.1 to 4.4 and their proofs are valid as they stand if G = Zm (m = ps, s ^ 1 ) and is assumed to act freely outside F. The statement 4.4 is of course not generally true for p = 2. However we have the following result: Let us keep the notation of 4.3,but assume that G = with m = 2s, s ^ 2 and that G acts freely outside F. Then n - r is even.

2^

Assume first F to be empty, n > 1. We want to show that the assumption n even leads to a contradiction. Since F is empty, we must have E2 ^ E^, by 4.1, hence, in thenotation of 2.1 and 4.3 (ii) dn+1(1 ® z ) = abt ® 1

(t = n/2)

.

It follows readily that H /Tp2k,nN _ -,2k+n+l,o dn+l(En+l } " n+1

H ri?2k+l,nx _ n dn-H(En+1 J" 0

and consequently that -g2k,n _ -g2k+1+n,o _ 00 00

q#

3

-g2k+1,n _ ^2k+n,o _ ^ OO 00 2

hence H1^ )

= Z2

but this contradicts 4.1 and 4.3*

(i ^ n + 1)

,

58

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CHAPTER IV

If F is not empty and n ^ 1, the proof of 4.5 is identical that of 4.4. Finally if n = o, then the square of a generator of G must necessarily leave each component of X invariant, hence (4.3, 4.5) its fixed point set is a homology o-sphere. Since by assumption it is equal to F, we have that n - r = 0 is also even in this case.

to

4.6. In4.7 the following properties of Steenrod’s operation Sq1 :H^(X; Z2 ) -------------- > (X; Z2 ) will be used: it is zero on the reduction mod 2 of integral cohomology classes; if Hi+1(X; Z) is a group of type (2, 2, ..., 2), then Sq1 defines a 1 1 +1 homomorphism of H (X; Z2 ) onto H (X; Z ) ® Z 2, hence in particular Sq1 is zero on Hi(Z2; Zg ) If and only if i is even (see 2.1); if X = A x B and c = a ® b with a e H*(A; Z2 ), b e H*(B; Zg ), then Sq1(c) = Sq1a ® b + a ® S q 1b.

it commutes with ma

4.7. We assume here that X is a compact cohomology n-sphere mod 2, whoseintegral cohomology group is finitely generated, and that p = 2 . Let be the torsion subgroup of H^(X; Z) and T the sum of the T-p The universal coefficient formula implies that ord T is odd, H^X; Z) = T± (i 4 0 , n), H^Xj Z)/Tn = Z, H°(X;Z) =Z. On Hn (X; Z)/TR, the involution g acts either trivially or by z -- > - z. We say accordingly that g preserves or reverses the orientation. By 4.3, F is a cohomology r-sphere mod 2 (— 1 < r £n). THEOREM (Liao [8]). n - r is even if and only if preserves the orientation of X(n > 1 ).

g

Let F be non-empty. Then H*(F^; Zg ) = H*(Z2; Z2 )(g)H*(Sr; Z2 ). Sq1 is zero on Hr (Sp; Z2 ), since it increases degree by 1, and on H°(Sr; Z2 ), which is the reduction mod 2 of H°(Sr; Z). It follows (4.6) that for i even Sq1 is zero on H^(Fr;Z0 ) if and only if r is even. Since Sq commutes with H (X^) --> H (Fq.) it will then be enough to show (*) Let F be non-empty. Then Sq1 is zero on H^tX^; Z2 ) if and only if either n is even and Gpreserves the orientation or n is odd and G reverses the orientation. If F is empty, then n iseven if and only If G reverses the orientation. Since T has odd order, we have H^(Z2; T) = 0 for j ^ 1 by 2.1, hence the natural map H^*(Z2; H*(X; Z)) -- > H^*(Z2; H*(X; Z)/T) is an isomorphism for j > 1 by the exact cohomology sequence associated to 0 > T -- > H*(X; Z) -- > H*(X; fe)/T -- > 0. Using 2.1, we easily ob­ tain for the spectral sequence (Ep) of----- over the integers:

§1*

CHAPTER IV: E^

(1

’0

= Z2,

THE ACTION OP

E^ 2j’“ 1 '

0

= E^',k =

OR T1: GLOBAL THEOREMS 0

(j ^ i; l £ k £ n -

1)

59 ;

) E£J,t = 0 (r^ n + 1 ) ;

if

G- preserves the orientation then

(2) if (3)

E ^ j ’H = Z2, G

E^2j'_1'n = 0

(j*i)

;

reverses the orientation then E£2j>n = 0;

E ^ " 1'11 = Z2

Assume now that G preserves the orientation. If n is even, then H2^+1(XqJ Z) = 0 (2i ^ n) since has no non-zero terra of total degree 2i + 1, therefore H^CX^; Z2 ) = H2i(X^; Z) (g)Z2 (2i ^ n+ 1) and is annihilated by Sq1. If n is odd, then Z) is Z2 or 0 (i ^ n + 1) because by (i) and (2 ) the terms of total degree i in form a group isomorphic to Z2 . If moreover F is non-empty, then Z2 ) = Z2 )is two dimensional. The universal coefficient formula implies then that H^CX^; Z) = Z2 for all i ^ n + 1 , conse­ quently Sq1 does not annihilateH^CX^; Zg ). This proves (*) for F non-empty and G preserving the orientation. If G reverses the orienta­ tion the proof is quite similar and is left to the reader. Assume now F tobe empty. We then have H^(XqJ Z2 ) = 0 for 1 > n + 1. But in El all groups E*^*^ with p + q ^ n + 1 are finite d i ” 2 groups, therefore, in order to have H (X^; Z2 ) = 0 , we must have E ^ ' ^ = 0 for p + q ^ n + 1 . If G preserves the orientation then E^2J>° = z2 must be a boundary for some differential. Since for 2j ^ n + the only possible non-zero elements of total degree 2j - 1 have the fibre degree n, we must then have E^2^“n“1,n 4 0 and hence, by (2 ), 2j - n - 1 is even, that is, n is odd. If G reverses the orientation, one sees in the same way that n must be even, and this ends the proof of (*). ^.8. REMARK. In^.1 to it is not necessary to assume X to be locally compact. This is clear from the purely algebraic approach of [7], as is mentioned there. It could also be seen here using 3-8 and the fact that Kunneth’s rule is valid for F x Bq , because B^ may be assumed to be a compact manifold (see XVI, §5)*

(j £ 1)

BOREL

6o

§ 5•

CHAPTER IV

The action of the circle group

The identity component of the compact Lie group noted by G°o 5.1. LEMMA. * so is

If H*(Xj L) L).

G

will be de­

is of finite type, then

G° acts trivially on H*(X; L), therefore the term E2 of the spectral sequence of X -- > B is given by the Kunneth rule. Since G G A (x)B and Tor (A, B) are finitely generated if A, B are, it follows, using 2.3, that EQ d is of finite type, hence a fortiori Hc (Xpo; L) is of finite type. Let

N = G/G°.

It is a finite group.

By 3*3,

X _ is a prinG cipal N-bundle over X^. There is then (3 . 4 (a)) a spectral sequence leading to L) in which E2 = H (N; HQ(X Q; D) . Since the cohomology G of a finite group with coefficients in a finitely generated module is of finite type our result follows. REMARKc Taking the results of XVI, §§4, 5 into account, one sees easily that 5.1 and its proof remain valid in cohomology with closed supports, for a non-necessarily locally compact space X. 5.2. LEMMA Let dimTX be finite. Assume that * # H (Bp ; L) is trivial for x 4 F and that H (X; L) x is finitely generated (respectively vanishes in degrees > N). Then (a) the same is true for (b) the same is true for infinite.

H*(X‘;L ); -X-

Hc (F; L)

if

G

is

In the proof, the coefficients L are omitted. If G is finite ■#* see III, 5.2. Prom now on G will be infinite, hence (2o4), H (B^) contains non-zero free submodules in arbitrary high dimensions. We show first that H*(F) and H*(Xf) are finitely generated if H*(X) is. By *1 5.1,Hc(Xq) is of finite type, hence (3 .6 )Hc(Pq) is finitelygenerated for i > dim-j-X. By 2.4 and the Kunneth rule, each H^(F) isembedded in some H^(Fq ) for suitably large i, therefore H*(F) is finitely generated; it also follows that is of> finite type, hence, by the cohomology sequence, HQ((X - F)G ) is of finite type. Using the isomorphism H*((X - F)q ) = H*(X* - F) of 3-4 and dim-^X1

h J(P)

A

-- > #

h J+1(X

- P)

—

>

jr h £(X'

) -- > l£(F) —

h J+1(x

) -- > *

jr

> H*+1 (X' - F) -- >

h £+1 (X1) --

>

of cohomologysequences, together with H^(X) = 0 for i ^ N, shows readily that it is enough to prove that H^(Xf - F) = o for i N; for this, we consider the spectral sequence (E ) of * : X - F --- > X ’ - F. — 1 If y = jt(x), then it” (y) = G/Gv . By the assumption and 2.4, Gv ■A x is finite and L is a field of order prime toord G . Since G acts trivially on its own cohomology, it follows(III, 2.3) that the projection yields an isomorphism a__ : H (jt~ (y)) -- > H (G) which is G -- > G/G — 1 independent of the choice of x in iT (y). Moreover, the existence of a slice at x (see VII) implies that there is an equivariant retraction —.1 — 1 of a saturated neighborhood of (y) onto (y). It follows immediately that for any a € H (G), the map y -- > a (a) is a continuous cross J * 1 section of the sheaf defined over X ! - F by the groups H_(*“ (y)) (see c also XII, §2); hence this sheaf is constant, and we have ordinary co­ efficients in Eg. If k = dim G and s is the highest dimension for which HC(X! then the maximum argument nj(x« - F) . , 4, ------ cocycle — ---- gives H^(X' - P) = H^+lc(X - P). However, the above diagram shows that «* is surjective in dimensions ^ N, hence we must have s < N - k. G°

If now G is not connected, combine the result just obtained for with 2.k and III, 3*1*. 5.3. THEOREM. Let din^X be finite and H*(X; L) be finitely generated (respectively vanishing in degrees ^ N). Then the same is true (a) for

X/G and for

P if L = KQ, G

= Tk;

(b) for X/G if L = Kp, Z, G is a finite extension of a toral group, and the orbit structure is finite; (c) for F, either if L = and G p-group, or if L= K^, Z,G = T^, structure is finite.

is a finite and the orbit

62

CHAPTER IV

BOREL

If any of these assertions, for a given L, is true for a closed invariant subgroup N of G and for G/N, then it is true for G since F(G; X) = F(G/N; F(N; X)) and X/G = (X/N)/(G/N). It is there­ fore enough to prove the various assertions of 5*3 in the two cases G finite and G = T1. The former one being taken care of by (III, U.3 , 5 .2 ), we assume from now on that T = T 1 . Then Gx is finite for x F. If L = Kq, (a),(b), (c) follow from 2 .1 , 5 .2 . In the remaining cases, there are a finite number of distinct isotropy groups. Let N be a finite subgroup of G containing all Gv (x F). ByIII, 5 .2 , * HC (X/N; L) is finitely generated (respectively vanishes in degrees ^ N). Now, G/N is a circle group which operates on X/N freely outside the fixed point set (1 .2 ), therefore the cohomology groups of (X/N)/(G/N) and F(G/N; X/N) are finitely generated (respectively vanish in degrees ^ N) by 5 .2 . Since these two spaces are homeomorphic to X/G and F(G; X) (see (1 .2 )), this concludes the proof. 5.^. Let G = T 1 and dimQX be finite. i = a mod 2 and i > dimQ X, we have (1

} Ss.a(2 ) dltn Hc ^

V

* Ss=a(2 ) dlm ^

V

a = 0, 1 ,

Then, for

= dlm

V

and there is equality if and only if the spectral sequence over -- > bg is trivial. The first H^X^;KQ ) = H^(F and third terms in spectral sequence

inequality is due to Conner [5]• For i > x B^; KQ ) by 3*7 (c), and the equality of (1 ) follow from 2 . 2 and the Kunneth rule. (Ep ) of X^ -- > B^ over KQ we have

' KQ

of

n, we have the second In the

(2 )dim H^(Xq) = S E^"s,s < £ E^“s,s = 2 Hi_s(BG; KQ ) (g)H®(X; KQ )

whence the inequality of (1 ), and the equality if E 2 = Ew . Now if there is a non-zero differential in (E ), it is easily seen, using the multi* plicative structure of H (Bp; L), that there are elements of arbitrarily i-s s high degree on which some differential is non-zero, hence dim E^ 9 < dim E ^ s's for suitable i > n and s ^ o, and therefore we have a strict inequality in (1 ). 5.5

Let

G = Tk, and

^ ( 2 ) dlm ^

dimQX

be finite.

Ko } * Ss=a(2 ) dlm Hc(F’ V

Then (a = 0, 1 )

.

This follows from 5 A by induction on k. It will be proved in a later Chapter (XII), together with a similar result mod p, that if, under the present assumptions, the spectral sequence of Xq -- > B^ is trivial, and

§6

CHAPTER IV:

THE ACTION OF ^

dim H*(X, KQ ) is finite, then 5 .6 .

OR T 1 : GLOBAL THEOREMS

dim H*(X; KQ ) = dim H*(F; KQ ).

It follows in particular from 5*5 that if

so is

X

over

Q,

and and

5.7* Let X be compact and finite dimensional over KQ, X G = Tk . Then F is a cohomology r-sphere (- 1 < r < n) over n - r is even [5 ].

Is acyclic

F. Sn KQ

The proof is entirely analogous to that of the Smith's theorem (4.3), with r < n following now from 5 *3 , andthe dimensional parity from 5 .5 * 5 .8 . LEMMA. Let G = T1, din^X < H*(X; Z2 ) = H*(Sn; Z2 ), F = F(Z2; X), where Z2 is the subgroup of order two of G, and r be the integer such that F ~ 2 Sp. Then n - r is even.

For x i F-, Gy is finite of odd order, therefore, by 2 .1 , 3*6 * * * r : H (X^; Z2 ) -- > H (F1G; Z2 ) is an isomorphism in sufficiently high dimensions. By use of 2 . 2 we deduce immediately that the spectral se­ quences mod 2 of Xq -- > Bq and F ^ -- > Bq are simultaneously trivial or not trivial, and the lemma follows. 5-9» THEOREM. Let G = T^,' L = Z, Kp, X be compact, X ^ Sn, dim^X < 00• Assume moreover that G has only a finite number of distinct isotropy groups on X. Then F Sp and n - r is even. By induction, it is enough to consider the case k = 1 . For a given prime p, there exists an integer s such that F is the fixed point set of the subgroup of order ps of G. Thus for L = Kp, (p 4 0 ) our assertion follows from k.k if p ^ 2 , from 5 . 8 if p = 2 . For p = 0 we may apply 5 -7 * 5-3

In the case L = Z, the theorem is a consequence of the above, (c) and the universal coefficient theorem. §6 .

The quotient of a cohomology sphere by Z^ or T

The topic named in the title of thissection can be studied con­ veniently in the present framework. We content ourselves by giving some examples, without aiming at completeness. 6 .1 .

THEOREM [6 , 8 ].

Let

p

be a prime (respectively,

BOREL

64

CHAPTER IV

p = o). Let G = Zp (respectively, G = T1 ), X and diriipX < Let r be such that F ~ Sp. H*(X>; Kp) = Hi_r_1(BG; Kp) (r + 2 £ i < n), H^-(XT; Kp) = o otherwise for i / o .

SR Then

If F is empty we are essentially in the principal bundle case and our statement is well known. From now on P is not empty. Then A* : H*(Bq ) -- > H*(Xq ) is injective (3• 3 )• Since X ~ Sn, the spectral sequence of this fibering is then trivial and ( 1)

H^(Xq ) = H ^ B q ) + Hi - n (BG )

.

Consider now the spectral sequence of the map : Xq. -- > X 1. There E2 = H (X!; F) where F is the Leray sheaf of whose stalk at y e Y f is H*(*”1(y)). (See XVI, §§1, 2). Here *“1(y) may be identi­ fied with Bq (x €. jr"1(y)) by 3 . 1 . For y 4 F, its cohomology is .x trivial (2 .1 ), hence (q / 0 ) is concentrated on jt” (F). But it’1(P) = Fp = P x Br, therefore F^ is constant over F, and we get s, o = H (X1) E|*° 2 2)

(s

0)

= Hr (F; ^(Bq)) = ^(Bq)

(t 4 0 )

E°,t = H°(F; Et(BG )) = H^Bq)

(t 4 0 )

E^,1: = 0

(s 4 0 , r; t 4 0

On the other hand, we assert that (3)

K*(Hi(X' )) = 0

(i 2: 1 )

In fact, if (3 ) is wrong, then Im n contains some power of the polynomial generator of H (B^); (this is clear for p = 0 , 2; for p odd, it follows from H2i(Zp; Z^) = PH21-1 (Z^; Z^) where p is the Bockstein homomorphism) hence H^(X!) 0 for arbitrarily high j. However it must be zero for j ^ n; this last assertion follows from III, 4 . 3 or [2] if p 4 the argument of [2] is also valid for p = 0 . Using (1 ), (2 ), (3 ) and 3*4, it is then seen without difficulty that the differential di (2 £ i < n - r) induces an isomorphism of E?'1”1 = E^/1”1 onto E^+r'° = E2+r'° and that otherwise, the differentials are zero. This proves the theorem. In writing (2 ) we tacitly assumed r / 0 . If r = 0 we have E°*^ = H^Bq) + H^(Bq), Eg'^ = 0 (s, t 4 0 ) and the rest of the proof is the same.

4

6 .2 . LEMMA.

Let

G = T1, p

be a prime or zero,

§6

CHAPTER IV:

THE ACTION OF

OR T 1 : GLOBAL THEOREMS

dinip X < 00, X compact, X ^ Sn, and r be the in­ teger such that F 1 = F(Zp; X) is a cohomology r-sphere mod p. Then (k*3> 5 -8 ) n - r is even and H ^ X 1 - F.j; is equal tofor i = r + 1 , r + 3 , ..., n - 1 and is zero otherwise. We leave the case

n = r

to the reader and assume

65

)

n - r ^ 2.

Then (4) h£+1(X - F,; Kp) = h£(X - Fi5 Kp) = Kp; hJ(X - F,j by the cohomology sequence of order prime to p, therefore (5)

) = 0 (i 4 r + 1, n

X mod F 1 • For x i F1, Gx is of finite H (B^ ; Kp) is trivial (2 . 1 ) and

H*((X - F 1 )Q; Kp) = H*(X» - F\; Kp )

by 3*k-. Thus the spectral sequence of (X - F 1 )G -- > B^ leads from H*(Bq_)® H*(X - F 1 ) to H*(X! - Fj ). Since the latter vanishes in high dimensions, our assertion follows from (k) by an easy spectral sequence argument. 6 .3 . THEOREM [6 ]. Let G = T 1 have a finite number of distinct isotropy groups on the compact space X. Let L = Z, Kp. Assume dim-j-X < 00 and X ^ Sn « Let r be the integer, congruent to n mod 2 , such that F ^ Sp (see 5»9)« Then H ^ X 1; L) = L for i = 0 and i = r + 3 , r + 5 , ••, n - 1 and is zero other­ wise.

If L = KQ, see 6 .1 . If L = Kp (p prime) and, in the notation of 6 .2 , F = F-j, then use 6 . 2 and the cohomology sequence of X 1 mod F. If F ^ F^, proceed by induction on n, apply the hypothesis to F 1 and use again 6 . 2 and the cohomology sequence. If L = Z, the result follows from this and from 5*3 (b ). BIBLIOGRAPHY [1 ]

A. Borel, "Sur la cohomologie des espaces fibres principaux et des espaces homog£nes de groupes de Lie compacts", Ann. Math. 5 7 (1953), 115-207.

[2 ] A. Borel, "Nouvelle demonstration d fun th£oreme de P. A. Smith", Comm. M. H._29(1955), pp. 27-39. [3]

G. Bredon, "Orientation in generalized manifolds and applications to

66

BOREL

CHAPTER IV

the theory of transformation groups", to appear. [4]

H. Cartan and S. Eilenberg, Homological algebra, Princeton 1 9 5 6 .

[5]

P* Conner, "On the action of the circle group", Michigan Math. Jour. (1957), 241-7.

[6 ]

P. Conner and E. E. Floyd, "Orbit spaces of circle groups of transformations", Annals of Math. 67 (1 9 5 8 ), 90 -9 8 .

[7]

A. Grothendieck, "Sur quelques points d'algebre homologique", Tohoku Math. Jour. 9 (1957), pp. 1 1 9 -2 2 1 .

[8 ]

S. D. Liao, "A theorem on periodic transformations of homology spheres", Ann. Math. _56 (1 9 5 2 ), pp. 68 -8 3 .

[9]

N. Steenrod, Topology of fibre bundles, Princeton 1 9 5 2 .

[1 0 ]

C. T. Yang, "Transformation groups on a homological manifold", Trans. A. M. S._j7 (1 9 5 8 ), pp. 2 6 1 -2 8 3 .

V:

THE ACTION OP

OR

T1: LOCAL THEOREMS

A. Borel § 1.

Conservation of cohomological local connectedness 1.1.

The following elementary remark will be used in the sequel.

Let A'

I

A"

f

I '

! -- > B B -- > B 1 u

V C —

J8 V > C1

be a commutative diagram of modules and homomorphisms, with exact middle row. If v 1 o u 1 and v 0 u have zero (respectively, finitely generated) images, then so does g 0 f. 1.2. LEMMA. Let U = UQ ) U1 ) U 2 = V be closed invariant subsets of X such that Im r TT TT i i+1 (i = o, 1 ) is of finite type. Then Im r*TT v is of finite type. G G Let G° be the identity component of G. We prove first that ■X* Im rcU y is of finite type. Let (j_Er ) be the spectral sequence over G° G° L of U Q -- > G Q, and r^ ^+1 the homomorphism (j_^r ) -- > iG G * defined by inclusion. Since G is connected, we have ordinary coefficients in ^E2, hence there are exact sequences

0 —» H^B 0;L) ® h £(U1;L)-» ±E^ 3 -» Tor(Hr+1 (B q ;L), H^tL^L))-* 0 G

G

67

68

CHAPTER V

BOREL

compatible with the maps

r._L,

J_+ I

Since

-X

H (Bq U : L)

is of finite type,

the maps of the ® and Tor terms have finitely generated images, hence 5r's _ .i+2E2 Ep's has a finitely generated image (1.1 ). It is then true -_ - >^ i+2 2 2 a fortiori for theE 00 terms, and our contention follows. y U

is a regular covering of U1n, with group N = G/G° (IV, IG 3 .2 ). In the Cartan-Leray spectral sequence of this covering, ^E2 = H*(N; H*(U Q; L)), and there is again a natural map (j_Er ) -- > iG Since the cohomology of a finite group with respect to a finitely generated module is of finite type, Hr (N; H^(U Q; L)) -- > Hr (N; H^(V Q; L)) has G G a finitely generated image, and the lemma follows as above. 1.3. LEMMA. Let U = U0 ) U] )U2 = V be invariant open subsets of X. If Im n is of finite type i i+1 (i = 0 , 1) then Im . is of finite type.

Vo

The proof is the same as that of 1.2, except for a reversing of the arrows, and is left to the reader. 1.4. PROPOSITION. Let X be finite dimensional and clc over H (Bq ; L) is trivial for all x pointxset F. Let L = K^, Z and if L = Z. Then (a) X/G is clc-L° (b) If moreover H (B^; L) is is clc-j^.

a G-space, which is L. Assume that outside the fixed G be infinite

not trivial,

Coefficients L are omitted in the proof. We show first that if H (Bq ) Is also trivial or if F is empty, then X/G is clc^, using Wilder5s property (P, Q) (see I, 2.2). Let Q !, P 1 be open in X 1, with Q,1 compact and in P 1; there clearly exists R ! open In X 1, with Q 1 in R 1 and R ! compact In P 1. Let P, Q, R be their inverse images in X. Then Q and R are compact and contained in R and P respectively. Since X has property (P, Q), Imand Im j*R are finitely generated. By 1.3, Im is then of finite type. However, since

H*(Bq ) is trivial for all

x

x, jt1 : P^ -- > P 1 and

-- > Q T

*

induce Isomorphisms for cohomology (IV, 3A), hence Im jp , is finitely generated, and X ! has property (P, Q). This proves (a) for the case in which H (Bp) is trivial and also proves, In case (b) that X 1 - F is clc^« Assume now that H (Bq ) is non-trivial that F is not empty, and that G is infinite if L = Z.It remains to show that F is clc^ and that X 1 Is clc^ around a point of F.

§1

CHAPTER V:

THE ACTION OP 2^ OR T1: LOCAL THEOREMS

69

Let x € P. Since it has arbitrarily small invariant neighbor­ hoods, we can find by 1 . 3 closed compact invariant neighborhoods U ) V of x such that Im r-jjy and Im r^ y are of finite type® Consider now the commutative diagram ^ ^ h 1(u g ) —

vI

>

h *((u

n

p

)g )

I vr1

h ± (v g ) —

> ^ ( ( v n P)G )

where the maps are the natural restrictions. The horizontal ones are isomorphisms for i > dim^X by (IV, 3-6), therefore Im r1 is finitely generated. Now (U n P)Q = (U n P) x BQ, (V n F)G= (V n P) x BQ and the inclusion is the product of the identity on B^ and ofthe inclusion Y n F --> U n F. Given an integer s, there exists i > dim^X such that Kp"~s(BG ) contains a non-zero free L-module (IV, 2.4). It follows, using Kunneth !s rule, that Hs (V n F) and Hs (U n P) may be identified respectively with direct summands of H^((V n P)q.) and H^((U n P)q ) on which r1 induces the map yriF0 Therefore,Im i*unF ynF is finitely generated, and F is clc^ around X. It remains to show that X ’ is clcL around x. For this it is enough to prove: (*) Given an invariant neighborhood exists an invariant neighborhood V of x in is finitely generated. If W is a neighborhood of x of W and its transforms is an Invariant * r-jjy is trivial or finitely generated, so then find invariant closed neighborhoods

U U

in X of such that

x e F, there * Im pu /q ^y /q.

in U, then the intersection T neighborhood of x, and if -X* is Using the above we can U ) V ) W of x such that

rW ' rW ' rUnF,VnF, rVnp,WnF have flnltely generated images. If we then write down the three rows diagram whose rows are the exact cohomology sequences of U mod U n P, v mod V H P and W mod W n P, and whose vertical maps are the natural restriction maps we see then, by 1.1 that the image of Im Pc-g_p is finitely generated. Similarly, there ex­ ists a closed invariant neighborhood S of x in W such that Im rc¥_F S_F is finitely generated. By 1.2,. rQ : Hq((U - F)q ) -- > HC ((S - F)q ) has a finitely generated image. It follows then from IV, 3.4 that Im rcUj_F is finitely generated. From what has been proved so far, we deduce the existence of compact neighborhoods V, ¥ (U ) V )V) of x in X, invariant under G, such that rcu,np,Vinp have finitely generated images.

rcy-F,W'-F(U’ = U/G> V ' = V/G’ W ' = W/G) We then consider the commutative diagram

CHAPTER V

BOREL

TO

--- > H*(U'

- P) -- > H ^ U ' ) --- >

1

V

---> h £ (V '

v

---> h J ( V

n U' ) --- >

I

- P ) --> H ^(V ' ) --- >

N/

H^F

V

H ^F

n V ' ) --- >

V

- F ) --> H ^ V ) --- >

I

N/

H ^(F n V ) --- >

where the rows are exact cohomology sequences and the vertical maps are restriction maps. 1.1 then shows that r^t has a finitely generated image, and this proves (*). 1.5- THEOREM (Floyd [4]). Let X be a locally compact space which is finite dimensional and clc over Z, and let G be a finite group operating on X. Then X/G is clc over Z (and finite dimensional over Z by III, 5.1). First we consider the case where G = Z^. If y ^ F, then X 1 is clearly clc around y. Let now y e F. By the -universal coefficient formula, X is also clc over Z^, hence there exists by 1.4 compact invariant neighborhoods U ) V of y In X such that H*(V; Z) -------- > H*(W; Z) and H*(U/G; Z^) -- > H*(V/G; Zp ) have finitely generated images. We form the following commutative diagram in which the rows are the exact sequences defined by means of the exact coefficient sequence 0

> Z -£-> z --- > Zp --- > 0

and the vertical maps are the restriction maps:

H^U'; z) -- > v

(U 1; 2^) ---> \l

H1(V’; z) -- > Hi(Vl;

Zp ) ---->

v V

V

-- > H1 (W1; Z) -E-> ^ ( W j

j/ Z) -- > H1^'; Zp ) -- >

(U1 = U/G. etc.) We want to show that H^(U'; Z) -- > Z) has a finitely generated image. Since we already know this to be true for H1 (U*; Zp) -- > H^(V'; Zp ) it is enough (1.1 ) to show it for the composite

§1

CHAPTER V:

THE ACTION OP 7^ OR T1: LOCAL THEOREMS

map a = p = v : H^(V'; Z) -- > H^CW; Z). following diagram

H1 ^'; Z)

-- E-- >

71

To this effect we consider the

h^V'j

LV',W' V

Z)

v r V',W n (W; Z) *

V

u

7

H1 (W1; Z) ---2-- >

H^W'j Z)

Here the maps jt are transposes of the projections V -- > V' and — * W -- > W T, and n is the transfer map. We have p = |i o ^ by (III, 2.1 ), and the commutativity in the remaining parts of the diagram is clear. -> H (W; Z) has finitely generated image by construction, Since H^V; Z) H^V'; Z) -- > H (¥'; Z), and this ends the same is then true for If G is a finite p-group, the theorem the proof In the case G = follows then by Induction on ord G. nl nk , Let now G be a finite group of order n = p1 .°. pk (p^ prime, p^_ 4 Pj f°r I 4 J ) and let be a Sylow subgroup of order compact and Pj_ 0 < i < k). Let U ! ) V 1 be open in X/G, with V contained in U T, and U, V their inverse images in X. Then V (respectively V/K^) and U (respectively, U/K^) are openin X (respectively X/K^) and V (respectively V/K^) iscompact and con­ tained in U (respectively U/KL.) < k). In view of I, 2.2 and U/K. ), (1 < £ i £ * what has been proved previously, Im and Im JU/K^V/^ (1 < i < k) are y, is finitelyfinitely generated and it is enough to show that Im generated (integral cohomology). This follows from the existence of a commutative diagram HC (U'; Z)

-> z H^(U/K±; Z) A

H (V1; Z)

where

j

= 2 J-q/k. y/K 9 no

3um °^

t1*3113?03®3 °£ the projection

72

BOREL

maps (see III, proof of 5*2),and from (III, 5-2). 1.6. over

the fact that

CHAPTER V jc

is injective

THEOREM. Let X be clc and finite dimensional L. Then (a) If L = K0, and G = Tk, then X/G and F are (b) If L = Kpor Z, G is a finite extension of a toralgroup and the orbit structure is locally finite, then X/G is clc-^. (c) If L = Kp, G Is a finite p-group, or if L = Kp, or Z, G = T^ and the orbit structure is locally finite, then F is d c L.

If any of these assertions is true for a given L, for a closed invariant subgroup N of G and G/N, then it is true for G, since F(G; X) = F(G/N; F(N; X)) and X/G = (X/N/(G/N). It is therefore enough to prove the assertions of 1.6 in the cases G = T1 and G finite, and more specifically (a) for G = T1,(b) for G finite or G = T1 and (c) for G = Zp or G = T1. If G is finite, (b) follows from 1.5; if G = Zp, (c) follows from 1.4. Fronjf now on G = T1. Then Gx is finite for x £ F. If L = K , (a), (b), (c) follows then from 1.4 and IV, 2.1. In the remaining cases, the orbit structure is assumed to be locally finite; since clc is a local property, we may assume that G has only a finite number of dis­ tinct isotropy groups on X. Let Nbe a finite subgroup of G con­ taining all Gx for x i F. The space X/N is clcL by 1 .5 . The group G/N = T1 operates on X/N, freely outside its fixed point set, therefore F(G/N; X/N) and (X/N)/(G/N) are clc^ by 1.4. Since these spaces are homeomorphic toF(G; X) and X/G respectively, this concludes theproof. REMARK. When L = Zp and G is a p-group, the fact that is clc-^ is due to Smith [5, Theorem 6 .1 5 ], andthat fact that X1 clcL is due to Floyd [3,Theorem l1]. 1.7* The following be stated in a slightly more result of Conner [2] for the prove the cohomological local

F is

lemma willbe needed in VI. Itcould easily general form, so as to embody an analogous case G = T1, L = KQ, and be used above to connectedness of F.

Givena space A and a sub space B the map r ^ issaid trivial if there exists a point b e B suchthat Kerr ^ = Ker

to be

§1

CHAPTER V:

THE ACTION OF Z

OR T 1: LOCAL THEOREMS

73

LEMMA. Let X be a locally compact space which is of finite dimension n and clc over Zp. Let G = Zp operate on X, Fbe its fixed point set, and U = AQ ) A1 }••• 3 A* = V be a de­ creasing sequence of invariant compact subsets, with V connected, such that r* - . = rA A j-19 j is trivial for j = 0, ..., n - 1 Then dim Im runp^vnp £ 1• Let (jEp ) be the spectral sequence of the projection of KjQ onto Bp and r? - .the homomorphism of ( ...E ) into (,*E ), or of * ^ J— ',.J J” '-1 .J'j r r" J-UJ x ^ into H (A-^), the transpose of the inclusion map. In H xllou 11 v^jG particular = j_iE2 = H*(Zp; H*(Aj _1 )) — is the coefficient map defined by we have

0 )

(E3,t) = 0

*

r. - .. J” 1>J

(ti 1;

> H * ^ ; H*(Aj)) ^ ^ Since the latter is trivial,

0;

2; j t / Ts+1,t-1 jE- jJ V

(3 ) We have of course

jjS,t and (4)

r? - . is compatible with the isomorphism (3). J“ 19 J rj-i,j(j-ijl"3,3) C jJl_s+1,s'1

Therefore (1) implies

(s * i; j < n - i; i * o)

.

CHAPTER V

BOREL Since

rTT v GJ G

is the composition of the maps

r. - • it follows that

1, o

Im rUGVG C n the maP k. is zero in degrees ^ n and is an isomorphism in degree n. Therefore (O

= 0

(actually first for (2)

ft . j

^j

r = 2,

( t / n, r ^ 2 ;

but then a fortiori for

is an isomorphism of

-?Ep'n j

onto

r ^ 2),

i_iEp'n j

—■

j ^ 1)

and (j> 3

1) •

We now want to prove: (*) For j< n, jE2^n i 1)consists of permanent cocycles, so that .E^,n = _.E®,n (j < n; s

h£((f n U)G )

A

( 7)

=

hJ((f n u)

X

bg )

A

i1 = J J (PnV)G,(Fnu)G

J'uG ,vG hJ(vg )

-------- >

H^((F n v)G )

=

h£((f n v)

X

bg )

where the horizontal maps are given by IV, 3*^* Prom IV, 2.1 and the Kunneth rule we have for i ^ n, H^( (P n U)p ) = H*(F n U) and clearly, this isomorphism carries j over to jpnu FnV* For ^ ^ n + 19 horizontal maps are isomorphisms by IV, 3»5 and ji. ,r has a 1-dimensional * G* G image by 2«1« Hence Im j-pnu pny is 1-dimensional* This shows first (for x = y) that the sum of the local Betti numbers of P at x is 1. Therefore (I, 2 .2 ), P is clc; let k(x) be the dimension in which F has local Betti number 1 at x. Then it is easily seen that the set of x eF for which k(x) = k is open. Since F is locally connected, It followsthat itis aWilder manifold. Also, since the above discussion is valid for any y e F n U, we see by I, 3°^ that P is locally orientable. If now a given connected component FQ of F is contained In an orientable component of X, we may assume U n F = PQ, hence, again by I, 3*4, FQ is orientable. 2 .3 . THEOREMo We keep the assumptions of 2.2. Let k = di^p F0> n = dinip XQ, where FQ is a connected component PQ of F, and XQ the connected component XQ of X containing PQ. Then n - k is even in each of the following cases:

(a) (Bredon [1]), (b) p = 2 , G = Z^

p is odd. and G acts freely outside

78

CHAPTER V

BOREL

By induction again, it is enough to prove (a) for G = Z^. In the following we discuss simultaneously the case (b) and the case G = Z , p o d d . We have then (IV, 2 . 1 ): (8 )

Bq

H*(G; Zp) = Aa ® Zp[t>]

(see

(d°a = 1, d°b = 2 )

The operator defined by a on the cohomology XVI, §1 ) will be denoted by a.

.

of a fibering

over

Let x €F Q , U an open connected orientable neighborhood of x in X such that U n F is connected, hence equal to U n F , and orientableo We then can easily find a sequence (U.) of neighborhoods of x satisfying the assumptions of 2 . 1 , such that V n P is connected. As was seen in the proof of 2 . 2 V n F is then adapted to U n P, therefore the image of j 1 : H^((F n V)q) --- > H^((F n U)G ) is equal to h£(F n u) ® Then (8) implies that a annihilates Im j1 if and only if i - r is odd. On the other hand, in the notation of the proof of 2. 1 n E i-n,n . nE|-n ’n = H 1 - n ( G ) (g)H^(V) hence a(nE ^ - n , n ) = 0 if and only if i - n is odd. 2 . 3 follows now from 2 . 2 (7 ) and from the fact that in the diagram (7), all maps commute with a, and the horizontal homomorphisms are bijective. 2 .4 . Theorem 2 . 3 corresponds to (IV, 4.^). 2.^. 4.4). We now want to dis­ cuss an analogue for cohomology manifolds of IV, ^.7 4.7 when F is not empty. The proof parallels that of 2.3> with a replaced by S q 1• 1. Since it does not involve new ideas, it will be dealt with more briefly. Let X be an n - cm over Z, operated upon effectively by G = Z2 , with a non-empty set of fixed points F. Let x €F and U )V be invariant orientable connected neighborhoods of x in X. Then if G acts trivially (non-trivially) on H^(U; Z) = Z, G acts in the same way on H^(V; Z) since the isomorphism commutes with G. We say that G preserves (respectively r e v e r s e s ) the local orientation of X around x if it acts trivially (respectively non-trivially) on H^(U; Z) = Z. Here X is also a n - cm over Z2 , hence by 2 . 2 F is a generalized manifold over Zp , in particular It is locally connected. In the follow* * ing, j L will stand for the map j in cohomology with coefficients in L. LEMMA. Let ^ ^1 ^ ^ U2n a secluence °** invariant connected open subsets of X, orientable over Z, with U- - adapted to U.(j ^ 1 ), J

J

and

§2

CHAPTER V: let

THE ACTION OF Zp OR T 1 : LOCAL THEOREMS

U = UQ, V = Un . Then

79

Im j * ™ z = Z2,

1x0 j n v 1+7 = 0 (respectively Im j?Tk*ns7 = 0, UGVG' G G Ira = Z2 ), if G preserves (respectively G- G' reverses) the orientation of U. The operation Sq1 is zero on Im jE\+rn 7 (respectively (t Ct * p 2k+1+n ImjTT v 7 ) if G preserves (respectively reG Y 2 verses; the orientation. The first assertion follows from 2.1 and IV, 2.1. To prove the second assertion, let us consider the diagram 0

> H^(Vg; Z ) ® Z2 -- > Hg(Vgj Z2 ) -- > Tor(H^+1(VG; Z),Z2 ) -- > 0

0

>

i Jl h J(Ug ;


H^(UG; Zg ) -- > Tor(H;J+1 (UQ; Z),Z2 ) -- > 0

in which the rows are the exact universal coefficient sequences and the ■Xj.’s are defined by j . Assume the G preserves the orientation UGVG and let i = 2k + n. Then the first .assertion implies that = 0, henc Im j' consists of reduction mod 2 of integral classes and is therefore annihilated by Sq . One argues in the same way if G reverses the orientation. 2.5* THEOREM (Bredon [1]. Let G be a group of order 2 operating effectively on an n - cm over Z, with fixed points. Let FQ be a connected component of the fixed pointset of G, and r = dim2FQ. Then n - r iseven (respectively odd) if and only if G preserves (respectively reverses) the local orientation around some point of F0. COROLLARY. If G preserves the local orientation around one point of FQ, then it does so around every point of F . Let x e Fq. Using 2.1, 2.2 and^he lemma, we first find orientable connected neighborhoods U ) V of x in X with V adapted to U over Z, hence also over Zg , and U n F, V n F connected,

80

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CHAPTER V

orientable mod Z2, V n F adapted to U n F over Zg. Then the image of j1 : h J((V n F)0; Z2 ) -- > h J((U n F)g; Zg ) is Zg ) ® H^(F n U; Z2 ). Therefore Sq1 is zero on Im j1 if and only if i - r is odd. The theorem follows now from the lemma by using the diagram (7 ) in cohomology mod 2, noting that all maps commute with Sq1 and that the horizontal ones are isomorphisms for i ^ n + 1, by IV, 3 .7 . 2.6. The following result and will be used in a later lecture.

is due to C. T. Yang [6, Theorem

1]

THEOREM. Let X be a connected n - cm over Z, G be a finite group acting effectively on X and F the fixed point set. If G =f (e); then dim^F < n - 1 , and if the equality holds, then G = Z2. In this proof, F(H) denotes the fixed point set of asubgroup H of G. If H is a cyclic group of prime order q, then dim.pF(H) < n - 1 (by 2.2 and I, 4.6) hence dimzF(H) < n - 1 by I, 4.9 and dim^F < n- 1 since F is closed in F(H).In order to prove thesecond assertion, it suffices by I, 4.9 to show that if G 4 Z2, there exists a prime q and a subgroup H for which dim F(H) < n - 2. If the order of G is divisible by an odd prime, this follows from 2.4. Let now G be of order 2s (s ^ 2 ), and let x e F. The subgroup N of G which leaves the local orientation aroundx invariant is clearly of index .1 or 2 ,hence of order ^ 2 , and contains therefore a subgroup of G of order

2; for the latter we have then § 3.

dim2F(H)
n. hence H * ((U n F )Q; Z) = Ss s l ( 2 ) H ?(U n F ; Z ); the same Is true of course with U replaced by V; it i s is clear that these isomorphisms carry j into the sum of the J*unp vnp* therefore, Im JunF ynF contains a subgroup A 1 = Z (namely A^ for i = n(2)). If W is open invariant in V, then Im y , which is . G a priori contained in Im j-y y , is in factequal to it, since it is a G' G # direct summand by 2.1 . As a consequence A 1 C Im j^np WnF’ Clear,ly> i-f* W runs through a fundamental system of invariant neighborhoods of a point x € U n F , then W n F runs through a basic system of neighborhoods of x in F . Thereforewe have shown that for each x e U n F , there exists a subgroup A(x) of H (U n F ; Z ), isomorphic to Z, and contained * in Im j un;F W f>or e v e r3r °Pen neighborhood W of x in U n F . On the other hand, we already know that U n F , is an orientable cm over any field. Moreover, it will be shown in VI that the orbit structure of a torus operating on a cmz is locally finite. Assuming this, F is clcz by 1 . 6 . The fact that U n F is an r - cm over Z now follows from (I, 4.11 )• Now

(U n f )g = (U n F ) x Bg ,

REMARK. The local only in the case L = Z, at the argument for L = Kp, p G = 7,, based on 2.2 and I,

finiteness of the orbit structure was used the end of the preceding proof. In particular, prime, is a simple reduction to the case of 4.6 which did not need any assumption about

§1+

CHAPTER V:

THE ACTION OF

OR T1: LOCAL THEOREMS

83

the orbit type. We point this out because in VI, the proof of the local finiteness of the orbit type will make use of the above Theorem 3 .2 , in the case L = Kp, p prime §4,

Remarks on local groups of the quotient space

We make here some remarks about the local cohomology groups of X/G around a fixed point x, in the cases considered in this chapter. They are merely meant to point out further applications of the method used here, and make no pretense at giving full proofs. These groups are computed in [1 ]. Roughly speaking, if n and r are respectively the dimension of the components of X and F containing x, they are the same as the cohomology groups of the quotient S /G, when F(G; S ) is a cohomology r-sphere. Such results, and the analogous ones for the circle group, can be obtained by combining the methods used in IV, §6 and in the proof of 2 .1 . More specifically, one takes a decreasing sequence of adapted neighborhoods (L L ) of x such that the n F are also adapted and considers the homomorphisms of the spectral sequences of UiG -- > Bq and of -- > U^/G. This is however a somewhat clumsy procedure, chiefly because the homomorphisms go in the "wrong" direction, namely, the one which leads to projective limits. Therefore it is better to try to avail oneself of the localization device of XIII, which uses inductive limits. Using arguments similar to those of XIII, it is seen without difficulty that the limit spectral sequence at x x BQ of X^ -- > X ! is the same as the spectral sequence of Y -- > Y/G when Y is a cohomology (n- 1 )-sphere, and F(G; Y) a cohomology (r- 1 )-sphere, the groups H^CY1 ) taking the place of the local groups I^(Xf). Thus in particular the results of IV, §6 i yield the local groups I„(X). Of course, this result can be used only I if one knows how to relate the groups Ix (X* ) to the usual local cohomology groups. According to some work now in progress by Raymond, it seems that, under suitable assumptions for the underlying space M, (for instance, metrizable, clc), may identified with the (i+1 )-st local cohomology group of M at x, (using reduced cohomology for 1 = 0 ). This of course checks with, and in fact would yield, the results of Bredon mentioned above. BIBLIOGRAPHY [1 ] G. Bredon, "Orientation in generalized manifolds and applications to the theory of transformation groups", to appear.

84

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CHAPTER V

[2]

P. Conner, "The action of the circle group", Michigan J. M. 4(1947), 241-247.

[3]

E. E. Floyd, "Some retraction properties of the orbit decomposition spaces of periodic maps", Amer. J. M. 73(1950, 363-7*

[4]

E. E. Floyd, "Orbit spaces of finite transformation groups II", Duke Math. J. 22(1955), 33-48.

[5]

P- A. Smith, "Transformations of finite period II", Annals of Math. 40 (1939), 6 9 0 - 7 1 1 .

[6]

C. T. Yang, "Transformation groups on a homological manifold", Trans. A. M. S. 8 7 (1 9 5 8 ), 2 6 1 -2 8 3 .

VI:

ISOTROPY SUBGROUPS OP TORAL GROUPS E. E. Ployd § 1. Introduction

The purpose here is to prove the following: 1.1. THEOREM. Suppose a toral group G acts on an n - cm X over Z. If C is a compact subset of X, then there are only a finite number of isotropy subgroups Gx, x e C. For differentiable actions, this is a special case of a theorem of Yang [8] (see also VII and VIII). For general actions, this has been proved for compact orientable manifolds by the author [3]; Mostow ex­ tended it to compact manifolds [6], as well as to compact Lie groups (see VII). In the above form, the theorem has been proved by Larry Mann [k]. Theorem 1.1 will be obtained as a corollary of which will be proved in Section 4.

(1.2) below,

1.2. THEOREM. Suppose the compact abelian Lie group G acts on the n - cm X over Z, and that {G^J is a sequence of closed subgroups of G with lim G^ = G. If C is a compact subset of X and if F(G^), F(G) denote the stationary points of G^, G respective­ ly then F(Gj_) n C = F(G) n C for i sufficiently large. We show now that (1.2) implies (1.1 ). Suppose (1.2) holds but that C is a compact subset of X with {Gx : x € C} infinite. There is a sequence {G„ } of distinct isotropy subgroups, x. e C. The i 1 collection of all closed subsets of the compact metric space G is itself a compact metric space, using the Hausdorff metric. Hence we may as well suppose, taking a subsequence, that the sequence {Gv } converges in the i 85

86

FLOYD

CHAPTER VI

Hausdorff metric to a closed subgroup H of G. Since almost all G„ are contained in each neighborhood of H, it follows from a theorem 1 of Montgomery and Zippin [5] that almost all G__ are conjugate under i inner automorphisms to subgroups of H. Since G is abelian, then almost all G„ C H. Hence {G__ } is a sequence of closed subgroups of H with i I lim G„ = H. Now F(GV ) fl C = F(H) n C for i large by (1.2). Hence i i F(GV ) n C = F(GV ) n C for i, j large. Since x. e F(GV ) n C then xi xj 1 xi x.1 € F(G__ ) and G Xv . ( Gy . Similarly G„ C G„ and we get a contraX. x± x± X. diction. §2 . A regular convergence theorem We prove in this section a lemma which is the purely topological part of the proof of (1 .2 ). The reader may well decide to skip it tem­ porarily, and return to it after reading the later sections. 2.1. Suppose that X is a connected, orientable n - cm over L. Suppose also that A D A 1 are closed subsets of X such that there is a separa­ tion X - A 1 = U 1 U V' (U1 and V 1 open, non­ empty and disjoint) with neither U l nor V 1 con­ tained in A. Then 4 °* PROOF. There Is the separation X - A = U U V with U = U* n (X - A), V = V ! n (X-A). Consider the following diagram, with coefficients In L and each row the cohomology sequence of a pair: h £-1(a

) —

>

h £(u

v h £"1(a

) ©

V *)— — >

h £(u

j" h £(v

h £(x

)

v ) — H->

h £(x

)•

Itow and are onto. If c is a non-zero element of H^(X), there are elements c1, c2 in H^(U), H^(V) respectively with j^J(c1) = j^y(- c2 ) = c. Then jCc, © c2 ) = 0, and c1 © c2 = 5(c' ) for some c ! e Hg"1 (A). Then r(c’) 4 °> since jn(c1 © c2 ) 4 °« The remark follows. 2.2. NOTATION. Suppose that {Y^} is a sequence of closed subsets of the locally compact Hausdorff space X. Say that {Y^} con­ verges to the subset Y of X, written lim Y^ = Y, if every neighborhood

§2

CHAPTER VI:

ISOTROPY SUBGROUPS OP TORAL GROUPS

87

of a point of Y intersects all but a finite number of Y^, while every x Y has a neighborhood intersecting Y^ for only finitely many i. If lim Y^ = Y thenY u U Y^ is closed in X. We need now avariant of the regular convergence concept; for an exposition of regular convergence, see White [7]. Suppose lim Y^ = Y, and that {L^} is a sequence of co­ efficient rings (all principal ideal domains). Say that {Y^} converges regularly to Y over CL^} if given y € Y and a compact neighborhood A of y, there is a compact neighborhood B ( A of y and an integer I with the restrictions

H*(A n Y±; L± ) ---- >

H*(B n Y±; L± )

H*(A n Y; L± ) ---- >

H*(B n I; L± )

trivial for forall all ii^^II(using reduced

cohomology).

We useEyer’s Eger’sversion [1, p. 129 ] of a theorem of of the the author author in in the following. By a closed covering a of a space X we mean a finite collection of closed subsets of X which covers X. If A ) B are compact subsets subsetsof ofaaspace space X, X, and and if a, a, p are closed coverings coverings of of A, A, B respectively with withpp> a, a., writepn » oc (over L) if there there is is aa if projection nn :: pp -- > ot --> suchot thatsuch for that arbitrary forarbitrary elements elements B , ..., B,..., B^ B^ Of P, P, rJBon...njtBk,Bon...nBk r i B o n . . . n * B k , B o n . . . n B k 0 for for a11 all 1 i< (using reduced reduced co" coof == 0 ^ nn (using homology). The theorem in question yields the following: if AA-1 -1 DD A A D ••• D A2n A 2 n arecompact compactsubsets subsetsofofa aspace space and if a_>1 « n aQ «« nn ... ... «« nnQQ::2n 2n (over (overL), L), whereis awhere closed Is a covering closedcovering covering of AAj^__,, then then in in the the diagram diagram (coefficients (coefficients in in L) L)

Hk (an ^ —> Hk («2n) Hk(a_1 ) — £_>> Hk( orn ) — li-> (a2n)

Hk (A_1 ) — -— >

— ^n-> Hk (A2n )

where the it’s are projections, the r ls are restrictions, and the are the natural maps, we have Image r C Image and Kernel D Kernel for all k < n. 2.3* THEOREM. Suppose that X is a locally compact Hausdorff space with n = dim^X < «>, and that {L^} is a sequence of coefficient rings. Suppose that {Y^} is a sequence of closed subsets of X which converges regularly over {L^} to the non-empty subset Y,

x.!s

88

FLOYD where and for large. Y^ = Y

CHAPTER VI

is a connected orientable - cm over Y is a connected orientable n^ - cm over every i. Then m^ = n^ for i sufficiently In particular, if Y^ ) Y for all i then for i sufficiently large.

PROOF. Consider first the case in which Y consists of a single point y, so that n^ = o for all i. Suppose the conclusion false; we may as well assume m^ > o for all i. If U is a neighborhood of y with U compact, then Y^( U for i large. For otherwise Y1 n (U - U) 4 ® for arbitrarily large I since Y^ is connected, so that we would have Y n (U - U) 4 if now A and B are compact neighborhoods of y then A n Y^ = B n Y^ = Y^ for i large. Then m. m. H (A n Y±; L± ) --- > H (B n Y±; L± ) mi is theidentity map of H (Y^; L^), which is non-trivial since Y^ is orientable. This contradicts regular convergence, so that m^ = ofor i large. Consider hereafter the case in which Y consists of more than one point. Let y e Y, and let U be an open set (in X) containing y with U compact and with Y Q U. Denote by C the set Y n(U - U). Cover C with a finite collection a_1 of compact sets in X whose in­ teriors cover C, and denote by A_1the union of the elements of Suppose a_1 so small that y \ A-1 and Y - U (Z A_1. Suppose also that (2.4)

if

Bq, ..., B^ e

then int BQ n ... n int B^ D C 4 (4o5)])« Then nerve a_1 = nerve

and

BQ n ... n B^ 4 $ This is easily satisfied (see [2, n Y = nerve n Y^ for i large.

For each y 1e C select an element B e a_1 containing y 1 in its interior. There is a neighborhood B^, C B of y ‘with H*(B n Y; L± ) -- > H*(By, n Y; L± ), H*(B n Y±; L± ) ---- > H*(Byl nY±; L± ) trivialfor i large. There is a finite collection P of Byi,s whose interiors cover C. There is then a star refinement aQ of p whose interiors cover C. We may also suppose aQ so small that (2.4) holds for it. Define AQ to be the union of the elements of aQ. It is then seen that aQ n Y^ n» a_1 n Y^ over L^ and aQ n Y n» a_1 n Y over L^ for i large (see [1, p. 132]).Moreover nerve otQ = nerve aQ n Y = nerve aQ n Y^ for i large. Continueto get a-1, aQ, ..., a2n; let A^ denote the of theelements of a^. Suppose the construction made so that

union

§3

CHAPTER VI:

ISOTROPY SUBGROUPS OF TORAL GROUPS

89

ak+l n Yi n>> ak n Yi' ak+l n Y Ii>> ak n Y over Li for 1 larSe * Since Ak contains C = Y n (U - U) in its interior, it follows from lim Y^_ = Y that Ak contains n (U - U) for i large. Hence for i large there is the separation of Y^ - A^ n separating it into its intersection with U and with X - U. Hence by (2.1), m.- 1 m.- 1 H 1 (A_1 n Y±; L± ) --- > H 1 (A^ n Y±j L± ) is non-trivial.

In the diagram

^(cr., n Y ±; L± ) — — > Hk (an n Y±; L± )Hk (a2n n Y±; L± )

Hk (A_1 n y±j Lj_) — — > Hk (An n Y1; L± )

(2.6)

n Y±; L± )

Image r C Image \2, Kernel jt1 } Kernel \2

for i large and k < n. Since r'r =( 0 for k = m^ - 1, the equations (2.6) imply that tc1 4 0 for k = m^ - 1 and i large. Since nerve = nerve n Y = nerve n Y^ for i large then co­ incides with Jt1 : Hk(o:n n Y; ) -- > Hk (a2n n Y; ). But there is a similar diagram to (2.5) with Y. replaced by Y and with (2.6) holding. m. - 1 Since jt1 4 °> then X2 4 0 and H (A* n Y; L^) 4 °> i large. Since A* n Y is a proper compact subset of Y, then n^ = dim^ Y ^ m^ (i large) by I, Theorem 4.3. mi = ni

A symmetric argument shows

m^ ^ n^,

^op 1 lapSe* §3* Two lemmas

3 .1 . Suppose that G, either a toral group or a p - group (p prime), acts on an (orientable) cm X over 2^. Then each component of F(G) is an (orientable) cm over Z^. This was proved in V, 3 .2 . 3 .2 . Suppose that G, either a p-group (p prime) or a toral group, acts on X, where dim^X< n < «> and each component of X is a cm over Z^. Suppose that AQ ) A 1 3 ••• }A(n+i)j is a sequence of compact invariant subsets of X with rA * = C 1^L+1 over Z^ for each i. Then

so that

90

FLOYD pF (G )n A 0,F (G )n A (n+1

CHAPTER VI " °

over

Zp

*

# rA A is trivial in dimension 0, A. is conAo 1 tained in a single component of A . [This uses the fact that every neighborhood of a component of a compact Hausdorff space contains an open and closed neighborhood.] Since A1 is invariant, the component of X containing A1 is also invariant. It follows that it is sufficient to prove the assertion in case X is connected; it is trivially true in case n = o. Suppose it has been proved In dimensions < n, and that G operates effectively. PROOF.

Since

Now G 3 Zp, and each component of E(Z^) is a cm over Zp of dimension < n. Note also that AQ+1 is contained in a single component A^ of An, and that A^ is invariant. Then AQ ) A 1 ... ) An_>1 5 A^ are invariant under Z. and rA A , ..., rA Al are trivial. By V (1.7), * p o 1 V-i^n rP(Zp )nA0,F(Zp )nA1!1 = °*

Hence

P(Zp) n A0 DP(Zp ) n V l

rP(Zp)nA0,P(Zp )nAn+1 = °*

DP(Zp) n Aa{n+1 } } ... )P(zp ) n An ,(n+1),

we see that each set is invariant under rW 7 InA )nn = °* F(Zp )nAk(n+1 )'P(Zt )nA(k+1 )(n+l )

induction,

Considering

G,

and that

Moreover

dim F(Z ) < n - 1. p p -

By

i>;(G)nVp(G)nA(n+i)j - o. § 4.

Proof of Theorem 1.2

Suppose now that X Is a cm over Z, that the compact abelian Lie group G acts on X and that {G^} is a sequence of closed subgroups of G with lim Gj_ = G. Let C be a compact subset of X. We must show that F(Gj_) n C = F(G) n C for i large. Clearly F(Gi ) )F(G). It may beseen that lim FCG^) =F(G). It is sufficient to prove that every x e F(G)has aneighborhood C with F(G^) n C = F(G) n C for i large. Every x 6 F(G) can be seen to have arbitrarily small connected invariant neighborhoods. In particular we may choose an orientable such neighborhood. Hence it is sufficient to prove the theorem for X orientable. For purposes of induction, we replace the condition that X be an orientable cm over Z by the following: (1 ) dim^X < oo,(2 ) every component of X is an orientable cm over Z^ for every prime p, (3 ) given x e X and a compact neighborhood A of x there is a compact neighborhood B( A of x with r^g = 0 for allcoefficientgroups Zp, p prime. Note that if X satisfies (1 ), (2 ), ( 3 ) and a toral group G acts on X, then F(G) has (1 ), (2 ), ( 3 )*

CHAPTER VI:

ISOTROPY SUBGROUPS OP TORAL GROUPS

91

For (1) is obvious, (2 )follows from 3-1 and (3 ) from 3*2. An orientable cm X over Z satisfies (2 ) by I 4.5. An easy argument using the universal coefficient theorem shows that (3 ) holds. We shall now prove the theorem in case G is a toral group and Gj_ is a subgroup of prime power order pni . Suppose x eF(G) and that A is a compact neighborhood of x. There is a sequence A ) AQ ) ... D A(n+1 )! n = dim^X, of compact neighborhoods of x with A^ invariant and ra a =0 for all coefficient groups Z , p prime. Prom 3-2, i i+1 p

(*.1 )

V ( G

1 )>A ( n + l) ,nP(G1 ) = 0, r ; onp (G hA (n+i )jnp(Q ) = o

with coefficients in

Z^

for all

A(n+i)i n F(G.) (A^n+1 ^tn

i.

This being

F(G))is contained in a

true in dimension

0,

single component of

F(G^)(F(G)), namely the component containingx. If M^(M) is the component of F(G^)(F(G)) containing x, it is then clear that it is sufficient to show = M for i large. Let now E^(E) be the union of all components of F(G^)(F(G)) not containing x. Since F(G^) and F(G) are clc and hence locally connected, E^ and E are closed. Since F(G)u UF(G^) is closed and since E^ is closed in F(G^) then P(G) U UE^ is closed. From the pre­ ceding paragraph, every point of M has a neighborhood not intersecting E^, i large. Hence E uUE^ is closed. Since F(G^), F(G), M, are all invariant, it followsthat E u U^E^ is invariant. Hencewe may re­ place X by its open invariant subset X - (E u U^E^). That is, we are reduced to the case in which F(G^) and F(G) have a single component, and M respectively. Now from (4.1) we get that {P(Gi )} converges regularly to F(G) over {Z^ }. Also F(Gj_) and F(G) are orientable over

Zp . Hence by (2 .3 ),

proved in case

G

= M

for

is toral and each

G^

i

large.

Hence the theorem is

is of prime power order.

We now prove the theorem in the general case by induction on dim G, noting first thatIt is obvious for dim G = 0. Suppose dim G = n > 0 and that the theorem holds for dimensions < n.

-

Since lim Gj_ = G, G^ has a subgroup H^ of prime power order with order H^ > °°. Select a subsequence of {H^} which converges to a subgroup H of G. We may as well suppose this the original sequence• By the argument of Section 1, H^C H for i large. Now dim H > 0 since ord H^ --- > ». Replacing H by its identity component Hc and H^ by H^ by H^ n H°, we may as well suppose H a toral group. Now

H^( H, lim H^ = H, H^

is of prime power order, and H

is

FLOYD

92

CHAPTER VI

a toral group of dim > o. By our earlier case, F(H^) n C = F(H) n C for i large. Also F(H) satisfies (1 ), (2 ), (3). We have that G/H acts on F(H) and that the set of stationary points of G/H acting on F(H) is precisely F(G). Let \ : G -- > G/H denote the quotient map. Then (x(G.j_)) is a sequence of subgroups of G/H with lim \(G^) = G/H. Hence, by induction, F(G/H, F(H) n C) = F(x(G± ), P(H) n C) for i large. But F(G± ) n C( FO^) n C = F(H) n C. Hence F(G± ) n C( F(x(G± ), F(H) n C) for i large. That is, n c C F(G) n 0 for i large. The opposite inclusion is obvious, and the theorem is proved. BIBLIOGRAPHY [1 ] E. Djer, Regular mappings and dimension, Ann. of Math. 6 7 (1 9 5 8 ),

l19-149. [2 ] E. E. Floyd, Closed coverings in Cech homology theory, Trans. Amer. Math. Soc. 84(1957), 319-339* [3]

E. E. Floyd, Orbits of torus groups operating on manifolds, Ann. of Math. 65(1957), 505-512.

[4]

L. Mann, Dissertation, University of Pennsylvania, 1 959•

[5]

D. Montgomery and L. Zippin, Topological transformation groups, New York.

[6 ] G. D. Mostow, On a conjecture of Montgomery, Ann. of Math. 65(1957), 513-516. [7]

P. A. White, Regular convergence, Bull. Amer. Math. Soc. 6 0 (1 954), 431-443.

[8 ] C. T. Yang, On a problem of Montgomery, Proc. Amer. Math. Soc. 8(1957), 255-257.

VII s FINITENESS OF NUMBER OF ORBIT TYPES G. E. Bredon § 1 . Preliminary Remarks In Chapter V E. E. Floyd proved the following result: 1.1. THEOREM. If M is a cm over Z and if T is a toral group acting on M, then, for any compact set A C M , there are at most a finite number of distinct isotropy subgroups Tp of T for p e A. For M compact and orientable this result was originally proved by Floyd in [1 ]. The above generalization is due to L. Mann [2 ]. Floyd’s original result was generalized by G. D. Mostow [5] to apply to compact manifolds M with a compact Lie group G of transformations, the result giving in this case that there are at most a finite number of non-conjugate isotropy subgroups of G. An examination of Mostow1s proof reveals that there is essentially no use of the theory of cohomology manifolds or of the theory of transformation groups which is not already contained in the statement of Floyd1s theorem. That is, the transition between these two theorems is entirely group theoretical, and therefore It should be possible to extend the above (local) theorem in exactly the same fashion. The purpose of the present chapter is to give this more general group theo­ retical formulation of Mostow!s result and its applications to the theory of transformation groups. Most of the proofs given here are contained in Mostow’s papers [k] and [5] and are only slightly altered to suit our present purposes. By the term subgroup (as distinct from analytic subgroup) we will always mean a closed subgroup. §2 . Statements of the main results Our main group theoretical result is 93

94

BREDON

CHAPTER VII

2 .1 . THEOREM. Let G be a compact Lie group and let be a class of subgroups of G which is closed with respect to conjugation in G and is such that the set {S n T; S €