Transformation Groups 9783110858372, 9783110097450

Table of contents :
Chapter I Foundations
1. Basic notions
2. General remarks. Examples
3. Elementary properties
4. Functorial properties
5. Differentiable manifolds. Tubes and slices
6. Families of subgroups
7. Equivariant maps
8. Bundles
9. Vector bundles
10. Orbit categories, fundamental groups, and coverings
11. Elementary algebra of transformation groups
Chapter II Algebraic Topology
1. Equivariant CW-complexes
2. Maps between complexes
3. Obstruction theory
4. The classification theorem of Hopf
5. Maps between complex representation spheres
6. Stable homotopy. Homology. Cohomology
7. Homology with families
8. The Burnside ring and stable homotopy
9. Bredon homology and Mackey functors
10. Homotopy representations
Chapter III Localization
1. Equivariant bundle cohomology
2. Cohomology of some classifying spaces
3. Localization
4. Applications of localization
5. Borel-Smith functions
6. Further results for cyclic groups. Applications
Chapter IV The Burnside Ring
1. Additive invariants
2. The Burnside ring
3. The space of subgroups
4. Prime ideals
5. Congruences
6. Finiteness theorems
7. Idempotent elements
8. Induction categories
9. Induction theory
10. The Burnside ring and localization
Bibliography
Further reading
Subject index and symbols
More symbols

Citation preview

de Gruyter Studies in Mathematics 8 Editors: Heinz Bauer · Peter Gabriel

Tammo torn Dieck

Transformation Groups

w

Walter de Gruyter G Berlin- New York 1987 DE

Author

Dr. Tammo torn Dieck Professor of Mathematics Georg-August-Universität Göttingen

Library of Congress Cataloging in Publication Data Dieck, Tammo torn Transformation groups. (De Gruyter studies in mathematics ; 8) Bibliography: p. Includes index. 1. Topological transformation groups. 2. Lie groups. J. Title. II. Series. QA613.7.D53 1987 514'.3 87-5450 ISBN 0-89925-029-7 (U.S.) CIP-Kurztitelaufnahme der Deutschen Bibliothek TomEKeck, Tammo: Transformation groups / Tammo torn Dieck. Berlin ; New York : de Gruyter, 1987. (De Gruyter studies in mathematics ; 8) ISBN 3-11-009745-1 ISBN 978-3-11-009745-0 NE: GT

© Copyright 1987 by Walter de Gruyter & Co., Berlin. AH rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means - nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typesetting and Printing: Tutte Druckerei GmbH, SalzwegPassau. Binding: Dieter Mikolai, Berlin.

Ich habe den Bau eingerichtet und er scheint wohlgelungen. Von außen ist eigentlich nur ein großes Loch sichtbar, dieses fuhrt aber in Wirklichkeit nirgends hin, schon nach ein paar Schritten stößt man auf natürliches festes Gestein. Freilich manche List ist so fein, daß sie sich selbst umbringt, das weiß ich besser als irgendwer sonst und es ist gewiß auch kühn, durch dieses Loch überhaupt auf die Möglichkeit aufmerksam zu machen, daß hier etwas Nachforschungswertes vorhanden ist. K.

Preface

This book introduces the reader to the theory of compact transformation groups. The theory of transformation groups deals with symmetries of mathematical objects. This viewpoint obviously needs some restriction. Thus I have chosen to concentrate on the algebraic topology of Lie transformation groups. An introduction is (by definition) not a complete presentation of a theory. Rather, methods and ideas are explained. Even for the purpose of an elementary introduction to the theory of transformation groups, at least two more books are necessary: Transformation groups on topological and smooth manifolds. Geometric representation theory. The topics of this book are organized into four Chapters. Chapter I presents the basic language. Chapter II is concerned with cell-complexes, homotopy theory and (co-)homology theory. Chapter III explains localization methods in equivariant cohomology. Chapter IV is centered around the Burnside ring and additive invariants. Each Chapter has its own introduction. A reference of the form 111(6.2) refers to Theorem (Definition, etc.) (6.2) of Chapter III. The Roman numeral is omitted whenever the reference concers the chapter where it appears. References to the bibliography at the end of the book have the form P.A. Smith [1938a], referring to author and year of publication, and, if necessary, are distinguished by additional letters a, b, and c. I have tried not to repeat material which is readily available in other textbooks: See the section "Further reading" at the end of this book. Since the theory of transformation groups is not a "primary theory", the reader is expected to be acquainted with basic algebraic topology. Occasionally, some knowledge about topological groups or representation theory might be helpful. There are numerous references to the literature to provide the reader with background material. The presentation varies in difficulty. Basic notions and results are developed in detail while more special applications may rely on references to the bibliography. Moreover, I have often mentioned topics which are related to the text, thus giving a guide to further reading. The bibliography is by no means complete. It contains only those items which I have used or referred to in the text. The numerous exercises vary in character: They ask the reader to verify statements used in the text. They give further information, sometimes with references to the literature. They present examples. They are genuine exercises.

VIII

Preface

Stefan Bauer and Peter Hauck read the manuscript. They suggested many mathematical and linguistical improvements. Wolfgang Luck helped with the proof-reading. I thank them for their generous and accurate work. Naturally, I would have liked to write the book in my mother tonfue. Jedoch ... Der Knecht wirft beide Arm empor, als wollt er sagen: „Laß doch, laß!" Göttingen, December 1986

Tammo torn Dieck

Contents

Chapter I Foundations

1

1. Basic notions 2. General remarks. Examples 3. Elementary properties 4. Functorial properties 5. Differentiable manifolds. Tubes and slices 6. Families of subgroups 7. Equivariant maps 8. Bundles 9. Vector bundles 10. Orbit categories, fundamental groups, and coverings 11. Elementary algebra of transformation groups

2 10 22 32 38 46 50 54 67 72 77

Chapter II Algebraic Topology

95

1. Equivariant CW-complexes 2. Maps between complexes 3. Obstruction theory 4. The classification theorem of Hopf 5. Maps between complex representation spheres 6. Stable homotopy. Homology. Cohomology 7. Homology with families 8. The Burnside ring and stable homotopy 9. Bredon homology and Mackey functors 10. Homotopy representations

95 104 Ill 122 133 139 150 155 160 167

Chapter HI Localization

177

1. Equivariant bundle cohomology 2. Cohomology of some classifying spaces 3. Localization 4. Applications of localization 5. Borel-Smith functions 6. Further results for cyclic groups. Applications

177 183 190 197 210 218

X

Contents

Chapter IV The Burnside Ring 1. Additive invariants 2. The Burnside 3. The space of subgroups 4. Prime ideals 5. Congruences 6. Finiteness theorems 7. Idempotent elements 8. Induction categories 9. Induction theory 10. The Burnside ring and localization Bibliography Further reading Subject index and symbols More symbols

227 ring

227 240 248 251 256 260 266 271 279 285 295 306 307 312

Chapter I: Foundations

The first chapter comprises the following sections. 1. The basic terminology related to a transformation group: Group action, orbit space, fixed point set, and equivariant map. The problem of turning the homeomorphism group of a space into a topological group. 2. A collection of examples from various branches of mathematics to indicate the role of symmetry considerations. Basic concepts such as representations and homogeneous spaces, which are used throughout the book. 3. Point set topology of transformation groups. Proper actions of locally compact groups. 4. Elementary material about changing the group: Restriction, additive and multiplicative induction. 5. Differentiable manifolds as a tool to prove the basic slice theorem. Qualitative results about differentiable manifolds, such as the principal orbit type theorem or finiteness of orbit types. 6. Classifying spaces for families of subgroups. They generalize the classifying spaces of bundle theory and are basic for the method of induction over orbit bundles. 7. Equivariant maps as non-equivariant sections. Inductive construction of equivariant maps. Equivariant fibrations. 8. Principal bundles with automorphism group and their classifying spaces. 9. Equivariant vector bundles: A rather short list of elementary properties. 10. Introduction to the combinatorial aspects of a transformation group. The universal covering of a transformation group as a functor. 11. Generalization of classical representation theory of modules over group rings to algebra in functor categories. Explanation of this viewpoint by describing projective modules and a few notions from algebraic K-theory. Although in some of the earlier sections we consider general groups acting on arbitrary topological spaces, the intention of the book is to study compact Lie group actions. So, if not obvious from the text or otherwise stated, the reader should assume that the group acting is a compact Lie group. Subgroups are supposed to be closed. The transformation groups, if necessary, should have closed isotropy groups and closed fixed point sets. Usually, spaces are assumed to be Hausdorff spaces.

2

Foundations

1. Basic notions.

Transformation groups describe (continuous) symmetries of geometric objects. This section introduces the basic notions and notations that will be used throughout the text. The next section describes examples to illustrate these concepts. Let G be a topological group and X a topological space. A left action of G on X (also, a left operation of G on X) is a continuous map (1.1)

ρ: GxJSf -> X

such that

(0 efe, ( ,X)) = Q(gh, x) for g,heG, xeX (1.2) (ii) ρ (e, x) = χ

for χ e X, e E G unit.

A left (/-space (also, a transformation group) is a pair (X, Q) consisting of a space X together with a left action ρ of G on X. We shall usually denote the G-space (Χ, ρ) just by its underlying topological space X. It is convenient to denote Q(g, x) by gx. The rules (1.2) then take the familiar form g(hx) = (gfi)x and ex = x. Occasionally, we call any set-mapping (1.1) satisfying (1.2) a left action and specify continuous action if necessary. A right action is a map X x G -> X, (x, g) ι—»· xg satisfying (xh)g = x(hg) and xe = x. If (x, g) \—* xg is a right action, then (x, g) ι—»· xg'1 is a left action. Usually, we work with left G-spaces and omit the word left. The left translation Lg: X -> X, x H* £jc by g is a homeomorphism of A' with inverse Lg- 1. This follows from the rules LgLH = Lgh, Le = id (A'), which are just reformulations of (1 .2). The map g ι—» Lfl is a homomorphism of G into the group of homeomorphisms of X. The action is called effective if the kernel of g ι-*· Lff is {e}. It is called trivial if the kernel is G itself. An action is called free if gx = x always implies g = e. Occasionally, we denote the left translation by lg. (1.3) Example. Let H be a subgroup of G. The group multiplication H x G -»· G, (Λ, g) i-> A# is a free left //-action. There is a similar right action. A group also acts on itself by conjugation G x G -» G, (gh) H·» ghg~l. If G is a topological group, then these actions are continuous. Let A' be a G-space. Then R = {(x,gx)\xe X, geG} is an equivalence relation on X. The set of equivalence classes X mod R is denoted by X/G. The quotient map q: X -> X/G is used to provide A7G with the quotient topology. This space is called the orbit space of the G-space X. The equivalence class of x e X is called the orbit Gx through x. A more systematic notation would be G\A" for the orbit space of a left action and X/G for the orbit space of a right action. There are a few situations where we use this notation. An action of G on X is called transitive if X consists of a single orbit.

Basic notions

3

(1.4) Example. The orbit space of the right action G x H -> G is G l H, the space of right cosets gH. The space of left cosets should be H\G. The map G x GIB -» G///, (g'.gtf) ι-, g's// is easily checked to be continuous. Any G-space G/H with this action is called a homogeneous space. If K c G is another subgroup, then we can again form a left coset space K\ (G/H), the space of double cosets. A subset F c Xof a G-space A' is called a fundamental domain of this G-space if F (=. X -> X/G\s bijective. A fundamental domain contains exactly one point from each orbit. Usually, there are many different fundamental domains, and the problem then is to choose one with particularly nice geometric properties. For each xeX, the set Gx = {geG\gx = x} is a subgroup of G. This subgroup is called the isotropy group of X or of the G-space X at x. We let Iso (X) denote the set of isotropy groups of X. From GflJC = gGxg~l it follows that Iso (A') consists of complete conjugacy classes of subgroups. Recall that subgroups H and K of G are called conjugate in G (notation H ~ K) if and only if there exists g e G such that H = g Kg'1. The conjugacy class of N is denoted by (//). We call H subconjugate to K if H is conjugate to a subgroup of K. Subconjugation defines a partial order on the set of conjugacy classes of subgroups. We write (H) < (K) if H is subconjugate to K. (1.5) Example. Let S(G) be the set of subgroups of G. We have an action (without continuity) G x S(G ) -> 5(G ), (g, H) N* gHg ~ 1 . The orbit of H is the conjugacy class (H). (1.6) Proposition. Let X be a G-space and xeX. The map G -> X, g t-» gx is constant on cosets gGx and induces an injective map qx: G/GX -» X whose image is the orbit through x. D We leave the verification to the reader. In general, qx is not a homeomorphism onto its image (compare section 3). We specify the following subsets of a G-space X (1.7)

XH={xH=(JXK, all Κ^Η,Κή=Η κ as the reader should check. If A' is a G-space and Υ c Xa subspace, we call Υ G-invariant or a G-subspace if for all g e G and y e Υ the relation gy e Υ holds. Then we have an induced continuous action G x Υ -»· Υ, (g, y) H-»· gy and Υ becomes a G-space. The orbits are the smallest G-subsets of a G-space; each G-subspace is a union of orbits. The orbit bundles are G-subspaces. Of particular importance is the orbit bundle X{e}, the largest subspace of X where the action is free. The complement X\XM is often called the singular set Xs of X. It is the union of the fixed point sets JT", H*{e}. Suppose X and Υ are G-spaces. A map /: X -* Υ is called a G-map or a G-equivariant map if for all g e G and χ e X the relation f(gx) = gf(x) holds. When dealing with G-spaces, G-maps are usually supposed to be continuous. A G-map /: X -» Υ induces a map

(1.11)

(1.12) f/G:X/G

-> Y/G, Gx i-> Gf(x)

between orbit spaces. If /is continuous, then//G is continuous. If/: X -» Υ is equivariant, then, for all χ e X, (1.13)

Gx^Gfx.

If in (1.13) equality holds for all xeX, then /is called isovariant. Of course, G-spaces and G-maps form a category. As usual in categories, we have the notion of an isomorphism between objects; the term G-homeomorphism is also used in this context. If A" is a G-space and Υ a /sf-space for topological groups G and K, then Χ χ Υ becomes a G x AT-space with action ((£, k), (x, y)) i-> (gx,ky). Similarly, this definition can be extended to an arbitrary number of factors. The category of G-spaces has products: Suppose (Xj\jeJ) is a family of G-spaces. We define a G-action on the topological product Π Χ} by

which is also called the diagonal action. The reader may check that this gives the categorical product. G-maps /o,/!: X -» Υ are called G-homotopic if there exists a continuous G-map, a G-homotopy from/0 to/!, F:A-x[0,l] -» Υ

Basic notions

5

such that F(x, 0) =f0(x), F(x, 1) =fi(x); here, [0,1] has the trivial G-action and Ar * [0,1] the diagonal action. Each map/ t : χ ι-»· F(x, t) is then a G-map. As usual, one shows that being G-homotopic is an equivalence relation and that one has a category of G-spaces and G-homotopy classes of mappings. We use the symbol [A', Y~\G for the set of G-homotopy classes of G-maps X -> Y. Furthermore, we write /0 ~ GfY if/ 0 and fi are G-homotopic. If AT is a subgroup of G and ρ: G χ Χ -* Χ a G-action, then we can restrict this action to K, thus obtaining the AT-space ies%X = resKX which is A'together with the action ρ\Κ* X. More generally, if φ: G' -» G is a continuous homomorphism, then ρ (φ x idx) = ρ' is a G'-action on X. We say that a left action G x X -»· A", (g, x) H->· gx and a right action Α" χ Κ -> Jf, (x, fc) H» xfc commute if for all g e G, x e A', fc e K the identity g(x&) = (gx)k holds. If G is a Lie group, M a smooth (= C°° differentiable) manifold, then a smooth action of G on A/ is a smooth map G x M -* M, (g, m) ι-» gw which is an action. A smooth manifold together with a smooth action is called a smooth or differentiable G-manifold. Occasionally, we need the purely algebraic aspect of a transformation group. Let G be a group. It can be considered as a topological group with discrete topology and as such it is called a discrete group. A G-set X is a space X with discrete topology together with an action of the discrete group G on A' or, equivalently, a set X together with a map G x X -*· X, (g, χ) ι-»· gx such that (gA)x = g (Ax), ex = x. Any G-space may be considered as a G-set by forgetting the topology on G and X. Giving a G-set X amounts to specifying a homomorphism from G into the permutation group of the set X. (1.14) Proposition. Let H and K be subgroups of G. (i) There exists a G-map G/H -» G/K if and only if H is conjugate to a subgroup ofK. (ii) If a e G, a~l Ha a K, then we obtain a G-map Ra. G/H -> G/K. gH (-H. gaK. (iii) Each G-map G/H -» G/K has the form Ra for suitable α ε G with α~ιΗα^Κ. (iv) Ra = Rb if and only ifab~leK. Proof. Let/: G/H -> G/K be equivariant. Choose α e G such that/(e//) = aK, ee G the identity. Equivariance yields aK=f(eH} =f(hH) = hf(eH) = haK for all h e H, hence a ~l Ha c K. Suppose a~J Ha c AT. If glH = g2H, then g t = g 2 /i for some Λ e // and Ac = a& for some fc e AT. Therefore, - g2 ΑαΑΓ = g2 akK = g2 aAT;

6

Foundations

hence Ra is well-defined. It is a G-map by construction. We have proved (i) and (ii) and leave the verification of (iii) and (iv) to the reader, α Let Η be a subgroup of G and N H or NGH the normalizer of H in G, i.e. N H = {ne G\n~l Hn = H}. The group H is a normal subgroup of N H and NH/H will generally be denoted by WH. A G-selfmap of G/H has the form R„: G/H -» G/H, gH H* gn// for n'1 Hn cr //. In general, such an element n need not be contained in NH. It belongs to NH precisely when Rn is a G-automorphism of G/H. We thus have an isomorphism W H -> Aut G (G///), n'1 H H-» K„ of W7/ onto the group of G-automorphisms of G///. The following is easily verified. (1.15) Proposition. The action G/H x WH -> G/H,

(gH, nH) H·

is a free action. D Let ρ : G x A r - ^ r b e a continuous action of G on X. Let Jf = Homeo(Ar) denote the group of homeomorphisms of X. The adjoint map ρ: G -»· «Jf, g ι—» Le is a homomorphism and ε: J^ x X -»· A", (/, χ) ι-» /(x) is an action. The question naturally arises whether there exist topologies on the set Jf such that assertions of the following type are true: (i) ρ is continuous. (ii) Jf is a topological group. (iii) ε is a continuous action. There exist different suggestions for defining topologies on tf (Arens [1946]). We shall only consider the compact-open-topology of Fox (see e.g. Kelley [1955], Ch.7; Bourbaki [1961], Ch.10, §3; Maunder [1970], Ch.6). We recall the definition and a few properties. Let X and Υ be topological spaces and Yx the set of continuous maps X -» Y. Let Κ -+f.

Basic notions

7

This map has the following properties: (1.18) (i) (ii) (iii) (iv)

If A' is a Hausdorff space, then α is continuous. If Υ is locally compact, then α is surjective. If X and Υ are Hausdorff spaces, then α is an embedding. If JSfand Fare Hausdorff spaces and Υ is locally compact, then α is a homeomorphism .

It is also important to consider the evaluation map ε: Υχ x X -> Y, (/, x) t-*/(x). To this end we recall that a space is called locally compact if each neighbourhood of a point contains a compact neighbourhood. From (1.18, ii) we obtain: (1.19) If X is locally compact, then ε is continuous. Composition of maps is continuous in the following case. (1.20) Let X and Υ be locally compact. Then Zr x Yx -> Zx, (g,f) i-> g o/is continuous. If we specialize to ρ: G x X -> X and define 3? = Homeo(Ar) c: Xx as a subspace, then we obtain (1.21) Proposition, (i) ρ is continuous. If X is locally compact and Q continuous, then Q is continuous. (ii) IfX is locally compact, then 3ex3f^3i (/· ·*) ·~* /(*) are continuous. D There remains the problem under which conditions 3tf -»· jdf, /i-»/"1 is continuous for a locally compact X, because then Jf will be a topological group in the CO- topology. (1.22) Proposition. Let X be a compact Hausdorff space. Then Homeo(J\T) is a topological group in the CO-topology. The proof of the continuity of / H-+ /"1 uses a reformulation of the COtopology. For simplicity, let Λ' be a compact Hausdorff space and Υ a uniform space with the induced topology being Hausdorff. Recall that a uniform structure is given by specifying a suitable collection U of sets U a Xx X containing the diagonal. For K c X compact, L/e U, and/0 e Yx, let N(f0, K, U) = {/e Yx | (/(x),/0 (x)) € U for all x e K} . Then:

8

Foundations

(1.23) The sets N(f0, K, t/) form a neighbourhood basis of /0 in the COtopology. For a proof see Bourbaki [1961], p. 43. If Υ is a compact Hausdorff space, there exists a unique uniform structure U inducing the given topology: The U e U are the neighbourhoods of the diagonal in A' x A". Using (1.23) and this uniform structure, a proof of (1.22) is now easy. Proof of (1.22). If/ 0 e JV, then/0 is uniformly continuous. Thus, given Ve U with (x, y) e V => (y, x) e V, there exists W such that (x, y)eW implies (/£"' (*), /o"' 00) e P. Let /e Jr satisfy (/0 (x), /(x)) e W for all xeX. Then (*,/o~ '/(*))e Pfor all jc e JT and, since/is bijective, (/"' (*),/„" ' (x)) e Γ for all xeX. Using (1.23), this shows continuity of/i-»/" 1 at/ 0 . D There exist locally compact spaces X such that