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Ergebnisse der Mathematik und ihrer Grenzgebiete Band 85 Herausgegeben von P. R. Halmos P. J. Hilton R. Remmert B. Sz6kefalvi-Nagy Unler Mitwirkung von L. V. Ahlfors R. Baer F. L. Bauer A. Dold J. L. Doob S. Eilenberg K. W Gruenberg M. Kneser G. H. Muller M. M. Postnikov B. Segre E. Sperner GeschiiftifUhrender Herausgeber P. J. Hilton

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WuYi Hsiang Cohon1ology Theory of Topological Transforn1ation Groups Springer-Verlag Berlin Heidelberg New York 1975

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Wu Yi Hsiang Department of Mathematics, University of California, Berkeley AMS Subject Classification (1970): 57 Exx ISBN-l3: 978-3-642-66054-2 e-ISBN-13: 978-3-642-66052-8 DOl: 10.1007/978-3-642-66052-8 Library of Congress Cataloging in Publication Data. Hsiang, Wu Vi, 1937-. Cohomology theory of topological transformation groups. (Ergebnisse der Mathematik und ihrer Grenzgebiete; Bd. 85). Bibliography: p. Includes index. 1. Transformation groups. 2. Topological groups. 3. Homology theory. I. Title. II. Series. QA613.7.H85. 514'.2. 75-5530. This work is su bject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically those of translation, reprinting, re-use of illustrations, broadcasting. reproduction by photocopying machine or similar means. and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1975. Softcover reprint of the hardcover 1s t edition 1975

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Introduction Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tlp by more general compact groups. It is usually rather straightforward to deduce similar fixed point theorems for actions of p-primary groups (or extensions of torus groups by p-primary groups) directly from the corresponding fixed point theorems for actions of the group 7l p • However, various efforts to extend such fixed point theorems beyond p-pi·imary groups (or extensions of torus by p-primary groups) all eventually wound up with puz- zling counter-examples [C 6, C 8, F 2, H 5]. On the other hand, if the group is an elementary p-group, (i e., 7l; or T'" if p =0), then a far-reaching generalization of the Smith fixed point theorem, valid for all finite dimensional, locally compact spaces, can be formulated and proved in the framework of equivariant cohomology. (cf. Theorem (IV. i), § i, Ch. IV). The basic setting for our approach using the cohomology theory in compact topological transformation groups is the following equivariant cohomology theory introduced by A. Borel [B 5]. Let G be a compact Lie group and let X be a given G-space. Then the equivariant cohomology of the G-space X is defined to be the ordinary cohomology of the total space XG of the universal bundle, X -+ XG -+ BG, with the given G-space as the typical fibre. The reasons for adopting such an equivariant cohomology theory in terms of the universal bundle con- struction are roughly the following: (i) Intuitively and heuristically, the complexity of the G-action on X will be reflected in the complexity of the associated universal bundle, X -+ XG -+ BG; and the classical characteristic class theory clearly demonstrates that cohomology

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VI Introd uction theory can then be used to detect the complexity of the bundle, which, in turn, also detects the complexity of the· G-action on X itself. Therefore, the above definition of equivariant cohomology simply formalizes and also generalizes the classical characteristic class theory to the study of the topology of general fibre bundles. (ii) From a technical standpoint, the above equivariant cohomology theory naturally and successfully brings together the modern theories of fibre bundles, spectral sequences and sheaves in a nice convenient way. Hence, it not only possesses all the convenient formal properties that one expects, but also is effec- tively computable. Basic properties of this equivariant cohomology theory as well as some fundamental general theorems such as the localization theorem of Borel-Atiyah- Segal type are formulated and proved in Chapter III. In Chapter IV, we shall proceed to investigate the relationship between the geometric structures of a given G-space X and the algebraic structures of its equivariant cohomology HJ(X). From the viewpoint of transformation groups, those structures which are usually summarized as the orbit structure are certainly the most important geometric structures of a given G-space X. Hence, it is almost imperative that one should investigate how much of the orbit structure of X can actually be determined from the algebraic structure of HJ (X). Examples of specific problems in this area are: How much of the cohomology structure of the fixed point set, H*(F), is determined by the algebraic structure of Ht(X)? Is it possible to give a criterion for the existence of fixed points purely in terms of HJ(X)? Suppose F=F(G,X)=0 (is empty), how does one determine the set of maximal isotropy subgroups from H~(X)? In the special case of elementary p-groups, we shall formulate various commutative algebraic invariants of HJ(X), which are, then, proved to be intimately related to the orbit structure of the G-space X. No- tice that there are general counter-examples for almost all non-p-primary groups which clearly indicate the non-existence of a general relationship between the orbit structure of X and the algebraic structure of Hi; (X). Such a sharp contrast of behaviors between transformations of elementary p-groups and transformations ofnon-p-primary groups is probably one of the most profound as well as fascinating facts in the cohomology theory of transformation groups. In retrospect, this also explains why torus groups play such a central role in the representation theory of compact connected Lie groups, which, after all, is concerned with the special case of linear transformation groups. Methodologically, one of the central themes in the approach of this book is that the cohomology theory of topological transformation groups can be developed roughly along the same lines as the classical linear representation theory of compact connected Lie groups. A concise exposition of the theory of compact Lie groups and their linear representations is given in Chapter II. In order to present the theory of linear transformation groups as a prototype of cohomology theory of topological transformation groups, we purposely adopt a rather geometric approach, in which the orbit structure of the adjoint action plays the central role. Moreover, it will be clear from such an exposition that the following two basic theorems constitute the foundation of linear representation theory of compact connected Lie groups:

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Introduction VII (i) Structural splitting theorem for linear tori actions: every complex linear representiation of a torus group always splits into the sum of one-dimensional representations. (ii) Maximal tori theorem of E. Cartan: the set of maximal tori forms a single conjugacy class and G= U{gTg- 1 ; gEG}. The first result classifies linear tori actions in terms of an extremely simple invariant called the weight system and the second result reduces the classification of linear actions of a compact connected Lie group G to the restricted actions of its maximal tori. Correspondingly, in the setting of the cohomology theory of topological transformation groups, the above structural splitting theorem for linear tori actions can be generalized into various structural splitting theorems of the equivariant cohomology (cf. Chapter IV), which can be considered as the generalized splitting principle in the geometric theory of generalized charateristic classes. Similar to the linear case, one may also combine the structural splitting theorems with the maximal tori theorem to define a (geometric) weight system for topological transformation groups. Such a program is carried out explicitly for the special cases of acyclic manifolds and cohomology spheres in Chapter V; and for cohomology projective spaces in Chapter VI. Although the geometric weight systems are no longer "complete invariants" for topological transformation groups, they nevertheless determine the cohomology aspects of orbit structures of the restricted actions of maximal tori and, hence, also the orbit structure of the original G-action to a great extent. In Chapter VII, we apply the cohomology method to study transformation groups on compact homogeneous spaces. Due to the fact that compact homo- geneous spaces encompass great varieties oftopological types and that the study of transformation groups on them is just getting started, there is an abundance of natural problems in this area, but, so far, only a small number of testing cases have been successfully settled and most of them are as yet unpublished. Therefore, results in this chapter are rather incomplete, and they should be considered only as beginning testing cases that serve to indicate the existence of interesting problems and deep results. In a paper soon to be published, I hope to give a more systematic account of the applications of the cohomology method to the study of trans- formation groups on compact homogeneous spaces. This book is based on a course given at the University of California, Berkeley. I am indebted to the participants of that course, especially to Dr. T. Skjelbred who helped to prepare a preliminary draft of Chapters III and IV. Berkeley, in Spring 1975 Wu Yi Hsiang

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Table of Contents Chapter I. Generalities on Compact Lie Groups and G-Spaces . 1 § 1. General Properties of Compact Topological Groups . . . 1 § 2. Generalities of Fibre Bundles and Free G-Spaces . . . . . 6 § 3. The Existence of Slice and its Consequences on General G-Space .... 10 § 4. General Theory of Compact Connected Lie Groups. . . . . . . . . 13 Chapter II. Structural and Classification Theory of Compact Lie Groups and Their Representations. . . . . . . . . 17 § 1. Orbit Structure of the Adjoint Action. . . . . . 17 § 2. Classification of Compact Connected Lie Groups. 23 § 3. Classification of Irreducible Representations. . . 30 Chapter III. An Equivariant Cohomology Theory Related to Fibre Bundle Theory. . . . . . . . . . . . . . . . . 33 § 1. The Construction of H~(X) and its Formal Properties. 33 § 2. Localization Theorem of Borel-Atiyah-Segal Type . . 39 Chapter IV. The Orbit Structure of a G-Space X and the Ideal Theoretical Invariants of H~(X). . . . . . . . . . . . . . . . . 43 § 1. Some Basic Fixed Point Theorems . . . . . . . . . . . . . . 43 § 2. Torsions of Equivariant Cohomology and F-Varieties of G-Spaces 54 § 3. A Splitting Theorem for Poincare Duality Spaces. . . . . . . . 65 Chapter V. The Splitting Principle and the Geometric Weight System of Topological Transformation Groups on Acyclic Cohomology Manifolds or Cohomology Spheres . . . . . . . . . . . . 70 § 1. The Splitting Principle and the Geometric Weight System for Actions on Acyclic Cohomology Manifolds . . . . . . . . . . . . . . . . . 71 § 2. Geometric Weight System and Orbit Structure. . . . . . . . . . . 75 § 3. Classification of Principal Orbit Types for Actions of Simple Compact Lie Groups on Acyclic Cohomology Manifolds. . . . . . . . . . . 83

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x Table of Contents § 4. Classification of Connected Principal Orbit Types for Actions of (General) Compact Connected Lie Groups on Acyclic Cohomology Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 91 § 5. A Basis Fixed Point Theorem . . . . . . . . . . . . . . . . . . 95 § 6. Low Dimensional Topological Representations of Compact Connected Lie Groups. . . . . . . . . . . . . . . . . . . . . 97 § 7. Concluding Remarks Related to Geometric Weight System ..... 101 Chapter VI. The Splitting Theorems and the Geometric Weight System of Topological Transformation Groups on Cohomology Projective Spaces ........................ 105 § 1. Transformation Groups on Cohomology Complex Projective Spaces . 106 § 2. Transformation Groups on Cohomology Quaternionic Projective Spaces ............................ 112 § 3. Structure Theorems for Actions of 7lp-Tori on 7lp-Cohomology Pro- jective Spaces (p Odd Primes) . . . . . . . . . . . . . . . . . . 118 § 4. Structure Theorems for Actions of 7l2-Tori on 7l2-Cohomology Pro- jective Spaces . . . . . . . . . . . . . . . . . . . . . . .. 121 Chapter VII. Transformation Groups on Compact Homogeneous Spaces 129 § 1. Topological Transformation Groups on Spaces of the Rational Homotopy Type of Product of Odd Spheres. . . . . 134 § 2. Degree of Symmetry of Compact Homogeneous Spaces 148 References . . 160 Subject Index . 164

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Chapter I. Generalities on Compact Lie Groups and G-Spaces This chapter will briefly review the general facts about compact topological groups, fibre bundles, topological G-spaces and compact Lie groups that are necessary for the subsequent development. Basic concepts and definitions will be adequately explained; and pro<?fs of some fundamental theorems will also be included whenever short clear cut proofs are available. § 1. General Properties of Compact Topological Groups Naturally, a topological group G consists of both a topological structure and ·a group structure which are compatible in the sense that the group structure is continuous with respect to the topological structure. More precisely, the multi- plication and inversion mappings are both continuous: G x G-+G, (gl,g2)>-+gl . g2; G-+G, g>-+g-l. Similarly, a Lie group (or rather a differentiable group) G consists of both a differentiable structure and a group structure which are compatible in the sense that the multiplication and inversion mappings are both differentiable. For many problems, the subclass of compact topological groups, or more specifi- cally, compact Lie groups plays ·an important role. In this section, we shall sum- marize the basic properties of compact topological groups: (A) Averaging and Haar Measure Obviously, a finite group G (with discrete topology) is a rather special example of compact topological group. Suppose qJ: G-+G 1(V) is a given representation which represents G as a group oflinear transformations on a vector space V. The following well known "averaging method" is a natural, simple-minded way to show the existence of an invariant inner product on V (with respect to the action of G). Given an arbitrary inner product <x,y) on V, it is clear that the following "averaged inner product" (x,y):

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2 Chapter I. Compact Lie Groups and G-Spaces is an invariant inner product on V with respect to the action of G. And this is also the only effective way, so far, to prove the complete reducibility of representations of finite groups. Next let us consider the group of unit complex numbers Sl = {ZE<C; Izi = 1}. TopologicalIy, it is a circle and it is one of the simplest example of compact topological group with an infinite number of elements. If we parametrize the circle group in the usual way, i.e. Sl={e21ti8;O~e~1}, and f is a continuous function on Sl, then the integration of f S6f(e)de is clearly a generalized "average value of f". Similarly, to any linear representation cp:Sl-'>GI(V) and a given inner product <x,y) on V, the folIowing "(generalized) averaged inner product" is again invariant with respect to the given action of Sl on V. (x,y) = S6 <e21ti8 • x, e21ti8 • y) de. As for a general compact group G, it is natural to blend together the above two kind of "averagings". Namely, for a finite subset A={aj}~G and a continuous function f(g), one defines the averaging of f over A to .be HeuristicalIy, it is reasonable to expect that {A ·f} wilI tend to a constant function, 1(f), as a limit when A becomes more and more dense in G. This is exactly the idea of Von Neumann in defining the (invariant) Haar-integral on a compact group G. We state the result as the folIowing theorem and refer to Pontrjgin's book "Topological Groups" for a detail proof. Theorem 1.1 (Existence and uniqueness of Haar integral). Let G be a given compact topological group and C(G) be the space of real-valued continuous functions of G. Then there exists a unique continuous linear functional I:C(G)-,>IR. satisfying (i) (left invariant): l(fa(g))=I(f) for all aEG, where fa (g) = f(ag). (ii) (positive andnormalized): f~O=I(f)~O, and 1(1)=1. The above invariant linear functional I can also be considered as the integral with respect to the invariant measure-the Haar measure-on G with total volume 1, i.e., 1(f)=SGf(g)dg. Corollary (1.1.1). Every complex (resp. real) representation of compact group G is equivalent to a unitary (resp. orthogonal) represenation. Corollary (1.1.2). Every complex (resp. real) representation cp ()[ a compact group G is completely reducible. Hence it decomposes uniquely into the direct sum of irreducible representations.

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