Lecture Notes in Mathematics

Lecture Notes in Mathematics

Edited by A. Dold, B. Eckmann and E Takens

Subseries: ~/L'~~ ~totTl/~15

1408

I IHI

Wolfgang LQck

Transformation Groups and Algebraic K-Theory

Springer-Verlag

Berlin Heidelberg NewYork London ParisTokyo Hong Kong

Wolfgang LQck

Mathematisches Institut, Universit~.t GSttingen

Bunsenstr. 3 - 5 , 3400 G6ttingen, Federal Republic of Germany

Mathematics Subject Classification (1980): 57SXX, 18F25,

57Q10, 57Q12,

ISBN 3-540-51846-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51846-0 Springer-Verlag NewYork Berlin Heidelberg

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© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany

Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr.

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The main goal of algebraic topology is the translation of problems and phenomena from geometry to algebra. In favourable cases we obtain a computable algebraic invariant which decides a given geometric question. A classical example is the classification of compact connected closed orientable surfaces by the genus.

This book is devoted to the connection between transformation groups and algebraic K- theory. We shall construct invariants such as the equivariant finiteness obstruction, Whitehead torsion, and Reidemeister torsion taking values Lnalgebraic K-groups. We de- fine injections or isomorphisms to the algebraic K-groups from groups such as finite- ness obstruction groups, Whitehead groups, representation rings, homotopy representa- tion groups or units of the Burnside ring. These are used to answer questions of the following type:

When is a finitely dominated G-CW-complex G-homotopy equivalent to a finite G-CW- complex? Under which conditions is a G-homotopy equivalence between finite G-CW-com- plexes simple? Is a given equivariant h-cobordism trivial? When are two semilinear G- discs G-diffeomorphic? Under which conditions are the unit spheres of two orthogonal G-representations G-diffeomorphic? When are two oriented G-homotopy representations oriented G-homotopy equivalent? Is a given oriented G-homotopy representation oriented G-homotopy equivalent to the unit sphere of a complex G-representation?

These questions will be treated in detail. They are related to the general problem of classifying group actions on manifolds. This problem and in particular its connections to algebraic K-theory are the basic motiviation for this book. We concentrate on deve- loping the algebra. The algebraic tools and techniques presented here have applications to G-manifolds besides the one to the questions above . They will not be worked out, as this would exceed the scope of this book, but are discussed in the comments.

Roughly speaking, most of the material of chapter ! can be found in the literature whereas chapters ~ and III mainly contain unpublished work. The study of modules over a category was initiated by Bredon [1967], where an equivariant obstruction theory for extending G-maps was established, and by tom Dieck [1981], where the equivariant finiteness obstruction and the diagonal product formula were studied for finite groups

and simply connected fixed point sets. The author wants to express his deep gratitude to Prof. Tammo tom Dieck for his encouragement and generous help.

The book is based on a course given by the author in the winter term 1986/87 and on the author's Habilitationsschrift, G6ttingen 1989.

The author thanks Christiane Gieseking and Margret Rose Schneider for typing the manus- cript.

We briefly summarize the main results and constructions.

0.i. Modules over a category.

Let F be a EI-category, i.e., F is a small category whose endomorphisms are iso- morphisms. A RF-module M is a contravariant functor F --~ R-MOD into the category of modules over the commutative ring R . The functor category MOD-RF of RF-modules is abelian. We reduce the study of RF-modules, their K-theory and homological algebra to the study of R[x]-modules for x ~ Ob F and R[x] the group ring R[Aut(x)]iby the Cofiltration Theorem 9.39. and Filtration Theorem 16.8. The Cofiltration resp. Fil- tration Theorem assigns to a projective RF-module P of finite tpye resp. RF-module M of finite length a natural cofiltration

--~ --~ --~ P = {0}

P = Pn Pn-I "'" o

resp. natural filtration

--~ M 1 --~ ... --~ M = M

{0} = M ° n

such that the kernel of P'I --* Pi-I resp. cokernel of M i --~ Mi+ 1 can be expressed in terms of R[x]-modules S P resp. Res M which themselves are naturally constructed

X X

from P resp. M . The Cofiltration Theorem implies the SPlitting Theorem 10.34.

for algebraic K-theory of RF-modules

K n ( R F ) = _ ~ K n ( R [ x ] ) x ~ i s F

w h e r e x runs over the set Is F of isomorphism classes of objects and n ~ Z .

As a special case we obtain the well-known splitting of the equivariant Whitehead group of a G-spacB. For finite F and R a field of characteristic 0 the Filtra-

tration Theorem gives a second Splitting Theorem 16.29. for algebraic K-theory of RF-modules. These two Splitting Theorems are related by a K-theoretic Moebius inversion 16.29. In geometry this corresponds to switching between the isovariant and equivariant setting, or between the two stratifications {X H I H c G} and {X H [ H c G} of a G- space X . Besides the K-theoretic application we also obtain a computation of Ext- groups EXTRF(M,N) n by a spectral sequence whose E2-term is given by Ext-groups over

the various group rings R[x] (see 17.18. and 17.28.). We introduce and study gene- ralized Swan homomorphisms in section 19.

The algebra of RF-modules for F the discrete fundamental category H/(G,X) (see 8.15.) of a G-space X is the main ingredient for constructing and computing certain alge- braic invariants of G-spaces and the K-groups in which they take values.

0.2. Invariants for G-spaces.

Here is a list of the most important invariants we will construct for G-spaces and G-maps.

name symbol value group defined for page

Euler characteristic xG(x> uG(x) finitely dominated i00, 278,

resp. G-space X 360

U(G)

multiplicative Euler characteristic

hx(X) II Q*/Z* finitely dominated

(H) G-space X

mx(X) II Q*/Z* special G-space X

(H)

hx(X)I/m ~(G) ~ finitely dominated G-

= 17[ Z/IGI* space X

(H)

hx(f)~m ~(G) e G-map between finitely dominated G-spaces

368

368

387

387

finiteness ob- struction

oG(x)

resp.

K (ZOrG)

o

278, 360

reduced finite-

ness obstruction

fi(x) wG(x)

Ko (ZH/(G,X)) finitely dominated G-space X

r esp.

K (ZOr G) o G

Wa (x)

278, 360

52

name symbol value group defined for page

(equivariant) G ( f ) Wh(ZH/(G,Y))

Whitehead torsion resp.

Wh(ZOrG)

G-homotopy equivalence of finite G-CW-complexes resp. G-manifolds f : X - - ~ Y

284, 360

~ e o ( f ) WhG (y) geo 68

isovariant Whitehead ~so(B,M,S) Wh~so(M) Isovariant h-cobordism 85

torsion (B,M,N)

Reidemeister pG(x) W~(~Or G) Finite G-CW-complex 362

torsion with round structure X

pG(M) Wh(BOrG) M a closed Riemannian G- 375 manifold satisfying 18.43.

p~L(M) KI(NG)Z/2 M a Riemannian G-manifold 376

KI(QOrG)

reduced Reidemeister p-[X} KI(Z(IGI~r~ G) Finitely dominated G- 363

torsion space X with round

structure

G KI(BG)Z/2 Riemannian G-manifold M 377 Poincar4 torsion ppD(M)

We compute the value groups in terms of algebraic K-groups of certain grouprings and state sum, product, diagonal product, join and restriction formulas. The reduced finiteness obstruction is the obstruction for a finitely dominated G-space X to be G-homotopy equivalent to a finite G-CW-complex. The Whitehead torsion is the obstruc- tion of a G-homotopy equivalence of finite G-CW-complexes to be simple. Both inva- riants are defined geometrically and algebraically and these two approaches are identi- fied by isomorphisms waG(x) --~ Ko(Z~/(G,X)) and Wh~eo(X) --~ Wh(ZH/(G,X)) . Cer- tain relations between these invariants are established. Roughly speaking, Whitehead torsion is the difference of Reidemeister torsion, the reduced Reidemeister torsion is a refinement of the finiteness obstruction.

0.3. Maps between geometric groups and K-groups.

We give a list of maps relating geometrically defined groups to algebraic K-groups.

They connnect geometry with algebraic K-theory. We denote injections by >--~ and

~

isomorphisms by ~% ~ :

: waG(x) ® uG(x) ~ Ko(ZH/(G,X)) 283

$ : waG(x) m • Ko(ZH/(G,X)) 283

: Wh eo(X) -- ~ Wh(ZH/(G,X)) 286

: Wh~so(X ) ~ ~ whG(M) 86

P

: whG(y×Tn+I) ~n+l = K?n(Y)geo ~--~--• KGn(ZH/(G,X)) 299

: A(G)* - - • Wh(ZOr G) 131

SW : C(G)* , KI(@OrG)/KI(Z(IGI)OrG ) 385

S W ° : Inv(G) >--• KI(~OrG)/KI(Z(IGI)OrG) 386

SW : C(G)* • K (ZOrG) 385

o

p~ : Rep~(G) >---* Wh(@OrG) G 373

p-G : V ~ ( G , D i m ) >--• KI(~OrG)/KI(Z(IGI)OrG) 401

p-G : v~V(g,Dim) >--• K(G) 404

0.4. Applications to geometry

We restate the Isovariant s-Cobordism Theorem 4.42. saying that isovariant h-cobordisms are classified by their isovariant Whitehead torsion. We relate the isovariant and equivariant setting by an homomorphism ~ : WhOso(M) --• whG(M) . Provided that the weak gap conditions 4.49. are satisfied, we show that ~ is injective and determine its image and thus get the Equivariant s-Cobordism Theorem 4.51. We give counter- examples to the Equivariant s-Cobordism Theorem 4.51. without the weak gap hypothesis in Example 4.56.

We prove for a finite group G of odd order that the transfer on K ° and Wh induced by the sphere bundle of a G-vector bundle vanishes under mild conditions (see 15.29.).

These transfer maps appear e.g. in the comparison of isovariant and equivariant White- head groups and in the involution defined on them by reversing h-cobordisms.

We construct an homomorphism p~ : Rep~(G) --• Wh(@OrG), IV] - - • pG(s(v ® V)) and prove injectivity in 18.38. Hence spheres of real G-representations are classified up to G-diffeomorphism by Reidemeister torsion. This reproves de Rham's theorem that two

G-representations are RG-isomorphic if and only if their spheres are G-diffeomeorphic.

A G-homotopy representation X is a finite-dimensional G-CW-complex such that X H = S n(H) holds for H c G . Given two G-homotopy representations X and Y with d i m X H = d i m y H for all H c G , we want to determine the set IX,Y] G of G- homotopy classes of G-maps between them. If we have choosen a coherent orientation, then

DEG : [X,Y] G ~ II Z , [f] ~ (deg fH)(H )

(H)

is an injection. We give in Theorem 20.38. a set of congruences describing the image of DEG and hence [X,Y] G which can be computed from the difference of the reduced Reidemeister torsion p-G(y) _ p-G(x ) by generalized Swan homomorphisms. In particular we get that G-homotopy representations are classified up to oriented G-homotopy equi- valence by an absolute invariant, the reduced Reidemeister torsion.

0.5. On the concept of the book.

We have tried to keep the book fairly self-contained. We give the definitions, results and proofs in full generality and illustrate them by examples. At the end of each section there is a comment where the material of the section is put into context with the work of other mathematicians, further applications are discussed and additional references are given. More information and results are contained in the exercises.

We advise the reader to at least read through them.

This expansive way of writing means that the sections contain much more material and results in much larger generality than needed for the following sections of specific applications. Therefore we have tried to give the reader, who is only interested in a specific question, the possibility to pick out a single section and read it without knowing the others. Here is some advice for such a reader.

The chapters II and III are independent of chapter I. If one is interested in the algebra only, one may skip chapter I completely.

In chapter I one may begin with one of the sections 3, 4, or 5 directly as they are independent of one another and sections 1 and 2 are quite elementary.

Nearly all notions and results are stated for Lie groups G and proper G-actions without any assumptions about the connectivity of the fixed point sets. The notational and technical difficulties decrease considerably if G is a finite group and the fixed point sets are empty or simply connected. In this case a summary of the in- variants defined for G-spaces in chapter II is given in section 18 including their basic properties. Moreover, section 8 is in this case of no importance, as everything takes place over the orbit category. In particular this restriction does no harm if one studies G-homotopy representations.

If one is interested in the finiteness obstruction and torsion only over the group ring resp. for the universal covering of a G-space without group action, one may directly begin with section Ii and 12 thinking of RF as RG, and similarly for the ma- terial about the Swan homomorphism for group rings and its liftin~ in section 19.

An experienced reader can start with section 18 without having looked at the pre- vious sections since the necessary input from them is reviewed in the beginning of section 18. Although section 20. makes use of section 18 and 19, section 20 can be read without knowing section 18 and 19 because only the formal properties of Reide- meister torsion and Swan homomorphism but not their explicit constructions are needed.

0. Introduction

Chapter I: Geometrically defined invariants 1

I. G-CW-complexes 6

2. Maps between G-CW-complexes 32

3. The geometric finiteness obstruction 47

4. The geometric Whitehead torslon 60

A. Geometric construction of Whitehead group and Whitehead torsion 60

B. Simple structures on G-spaces 74

C. Simpie structures on G-manifolds 81

D. Isovariant and equivariant s-cobo~dism theorems 83

5. The Euler characteristic 99

6. Universal functorial additive invariants 108

7. Product and restriction formulas 120

8. Lift-extensions and the (discrete) fundamental category 136

A. Lift extensions 136

B. The (discrete) fundamental category and the (discrete) 143 universal covering functor

Chapter II: Algebraically defined invariants 156

9. Modules over a category and a splitting of projectives 162

A. Elementary facts about RF-modules 162

B. The structure of projective RF-modules 170

i0. Algebraic K-theory of modules over a category and its splitting 180

A° The algebraic K-theory of RF-modules 180

B. Natural properties of the algebraic K-theory of RF-modules 184 C. The splitting of the algebraic K-theory of EF-modules 192

D. %he Bass-Heller-Swan decomposition 204

Ii, The algebraic finiteness obstruction 211

12, The algebraic torsion 236

13. The cellular chain complex 259

14. Comparison of geometry and algebra 277

A. The finiteness obstruction 278

B. The Whitehead torsion 284

C, Product and restriction formulas 289

D. Some conclusions 299

Chapter III: R F-modules and geometry 305

15. The transfer associated with a G-fibratlon 311

16. A second splitting 325

17. Homological algebra 339

18. Eeidemeister torsion 355

19.

20.

A. Review of modules over the orbit category Or G B. Review of invariants for R O r G - c h a i n complexes C. Review of invariants of G-spaces

D. Construction of equivariant Reidemeister torsion E. Basic properties of these invariants

F. Special G-CW-complexes

G. Reidemeister torsion for Riemannian G-manifolds Generalized Swan homomorphisms

Homotopy representations

A. Review of basic facts about G-homotopy representations B. Homotopy representation groups

C. Classification of G-homotopy representations by reduced Reidemeister torsion

D. The special case of a finite abelian group

Bibliography

Index

355 358 360 361 363 367 374 381 392 392 396 401

407

413

425

Symbols 437

G E O M E T R I C A L L Y D E F I N E D I N V A R I A N T S

Summary

In the first section we collect elementary facts about G-CW-complexes for G a to- pological group. We prove the Slice Theorem 1.37. and 1.38. and deal in Corollary 1.40 with path lifting along the projection p : X --~ X/G . In Theorem 1.23. we show for a G-CW-complex X that X is proper if and only if the isotropy group G x is com- pact for all x ~ X .

We prove the Equivariant Cellular Approximation Theorem 2.1. and the Equivariant Whitehead Theorem 2.4. in section 2. We give criterions for a G-space to have the G- homotopy type of a G-CW-complex in Corollary 2.8., Corollary 2.11. and Proposition 2.12. and examine G-push outs of G-maps and their connectivity in Lemma 2.13.

In section 3 we introduce the finiteness obstruction wG(y) ~ waG(y) of a finitely dominated G-space geometrically. We call Y finitely dominated if there is a finite G-CW-complex X and G-maps r : X --~ Y and i : Y ~ X satisfying r o i =G id.

Elements in the abelian group waG(y) are represented by G-maps f : X --~ Y with a finitely dominated G-space as source and Y as target. Addition is given on re- presentatives by the disjoint union. The zero element is represented by ~ --~ Y and wG(y) by id : Y --~ Y .

Theorem 3.2.

a) Let X be a finitely dominated G-space, Then X i_~s G-homotopy equivalent to a finite G-CW-complex if and only if wG(x) vanishes.

b) The finiteness obstruction is a G-homotopy Snvariant.

c) The finiteness obstruction is additive on G-push outs. D

A typical situation, where the finiteness obstruction comes in, is the following.

Suppose X is a finitely dominated G-space for which we want to construct a (compact smooth) G-manifold M with M =G X . As any such M is a finite G-CW-complex, the vanishing of wG(y) is a necessary condition. Often constructions of G-spaces give

In section 4 we extend the geometric construction of Whitehead group and Whitehead torsion due to Cohen [1973] and StScker [1970] to the equivariant setting following lllman [1974]. A G-homotopy equivalence f : X --~ Y between finite G-CW-eomplexes is called simple if it is G-homotopic to a composition of so called elementary ex- pansions and collapses. It determines an element G ( f ) , its (equivariant) Whitehead torsion, in the Whitehead ~roup WhG(y) by its mapping cylinder. Elements in whG(y) are represented by pairs of finite G-CW-complexes (X,Y) such that the inclusion Y --~ X is a G-homotopy equivalence. Addition is given by the G-push out along Y and the zero element by (Y,Y) .

Theorem 4.8.

a) A G-homotopy equivalence f : X --~ Y between finite G-CW-complexes is simple if and only if TG(f) vanishes.

b) f =G g => G ( f ) = G ( g ) c) G is additive on G-push outs.

d) TG(gof) = G ( g ) + g, G(f) o

Let f : X --~ Y be a G-homeomorphism of finite G-CW-complexes. If G is trivial, f is simple by Chapman [1973]. This is not true for non-trivial G in general (see Example 4.25. and 4.26.).

If G is a compact Lie group and M a (compact, smooth) G-manifold, we define a preferred simple structure on M (cf. lllman [1978], [1983], Matsumoto-Shiota [1987]). Hence for any G-homotopy equivalence f : M --~ N between G-manifolds its Whitehead torsion is defined. It vanishes if f is a G-diffeomorphism.

A cobordism (B,M,N) of G-manifolds is an isovariant h-cobordism resp. (equivariant) h-cobordism if the inclusions M --~ B and

valences resp. G-homotopy equivalences.

We introduce the isovariant Whitehead group torsion

N --~ B are isovariant G-homotopy equi-

~ s o ( B , M , N )

WhOso(M) and the isovariant Whitehead of an isovariant h-cobordism. We restate the Isovariant s-Co-

h-cobordisms over M up to G-diffeomorphism relative M , if dim MH/WH ~ 5 (i.e.

the dimension of any component of MH/WH is not smaller than 5) for all H E Iso M holds (see Browder-Quinn [1973]. Hauschild [1978], Rothenberg [1978]). We construct an homomorphism

4.43. ~(M) : WhOso(M) ~ whG(M)

satisfying ~(M)(T~so(B,M,N)) = ~G(M --~ B) (see Proposition 4.44.). This leads to (cf. Araki-Kawakubo [1988])

Theorem 4.51. The Equivariant s-Cobordism Theorem. Let M fying the weak gap condition 4.49. such that dim MH/WH ~ 5 Then

a) b)

be a G-manifold satis- holds for H 6 Iso M .

Any h-cobordism over M is an isovariant h-cobordism.

~(M) is splSt injective with a certain direct summand whG(M) c whG(M) as

image.

c) Wh~(M) classifies h-cobordisms over M up to G-diffeomorphism relative M

Because of this result equivariant Whitehead torsion is important for the classifi- cation of G-manifolds. It is in general much easier to handle with the equivariant Whitehead torsion than with the isovariant one. If one wants to show that two G-mani- folds M and N are G-diffeomorphic, the general strategy is to construct an h-co- bordism (B,M,N) by equivariant surgery and then apply Theorem 4.51. As an illustra- tion we mention the classification of semilinear discs M (i.e. G-manifolds M such that for H c G the pair (MH,sM H) is homotopy equivalent to (D k S k-l) for appropriate k ~ 0) due to Rothenberg [1978] in Theorem 4.55. Such M is classified by the ~G-isomorphism type of TM for x ~ M G and the Whitehead torsion of

x

STM --~ M \ int DTM up to G-diffeomorphism, provided that M satisfies the weak

x x

gap conditions 4.49. and dim MH/WH ~ 6 for H ~ Iso M . The assumption 4.49. in Theorem 4.51. is necessary (see Example 4.56.).

In section 5 we introduce the (equivariant) Euler characteristic xG(x) ~ uG(x) and show that it is a G-homotopy invariant and additive under G-push out in Theorem 5.4.

We define Euler ring Lie group.

U(G) = U G ({point}) and Burnside ring A(G) for G a compact

All the invariants above satisfy sum formulas for G-push outs and are G-homotopy invariant. It turns out that they can be characterized just by these two formal properties as universal functorial additive invariants (see Theorem 6.7,, 6.9., and

6.11.).

In section 7 we derive from this characterization a product formula 7.1., a restric- tion formula 7.25, and a diagonal product formula 7.26. for the finiteness obstruc- tion, Whitehead torsion and Euler characteristic in a simple geometric manner. This requires a careful analysis for the problem h~w to assign to the restriction res X of a finite G-CW-c0mplex X to a subgroup H of the compact Lie group G a H- simple structure (see 7.10o). This is a non-trivial question if G/H is not finite, see e.g. the case where X is a homogeneous G-space G/K . It is remarkable that the geometric description of the restriction formula for infinite G/H and of the dia- gonal product formula for infinite G are much easier than the algebraic oneewe will develop in section 14.

The diagonal product formula is the main ingredient in constructing an homomorphism

7.39. ~ : A(G)* • whG({point}) .

A unit in the Burnside ring represented by a G-self equivalence f : SV ---+ SV of the unit sphere of an orthogonal G-representation is sent by ~ to (I-#(SV))-I~G(f) . No G is known where ~ is non-trivial. This map appears in the study of G-homo- topy representations in section 20 and in equivariant surgery (see Dovermann-Rothen- berg [1988]).

If X is a finitely dominated G-space, X ~ S 1 is G-homotopy equivalent'to a finite G-CW-complex by the product formula. This can directly be seen from Mather's trick which is used to define a geometric Bass-Heller-Swan in~ection

7.34. ~ : waG(y) ~ whG(y x S I) .

It relates finiteness obstruction and Whitehead torsion (cf. Ferry [1981a],

(Definition 7.36., Theorem 7.38.). These appear for example as obstruction groups for equivariant transversality in Madsen-Rothenberg [1985a].

~

In section 8 we deal with lifting a G-action to a G-action on the universal covering which extends the ~l(X)-action and covers the G-action, where G is an extension

and G denote the components of the of ~I(X) and G (Theorem 8.1.). If G O o

identity, we study the square

8.8.

:x ~. :X/G

1 i

X ~ XIG

O

In particular the space X/Go is important as it is simply connected and carries the action of the discrete group ~ (G) = G/G (Lemma 8.9,). We will read off a

O O

lot of algebraic information from the ~o(G)-space X/Go . We derive from 8.8. an e x p l i c i t d e s c r i p t i o n of the k e r n e l and c o k e r n e l of ~I(X) - - ~ ~I(X/G) in Propo- sition 8.10. (cf. Armstrong [1983]).

We organize the book-keeping of the components of the various fixed point sets in- cluding their fundamental groups, universal coverings, WH-actions and diagrams corresponding to 8.8. All these data are codified in the discrete fundamental cate- gory H/(G,X) (Definition 8.28.). The notion of the cellular ZH/(G,X)-chain complex

(Definition 8~37.) is the main link between geometry and algebra. It is the compo- sition of the discrete universal covering functor X/ : H/(G,X) --~ {CW-complexes}

(Definition 8.30° ) and the functor "cellular chain complex". The algebra of modules over a category is modelled upon it.

We i n t r o d u c e the e q u i v a r i a n t v e r s i o n of a C W - c o m p l e x and c o l l e c t its m a i n p r o p e r t i e s . We w i l l d e a l also w i t h the case w h e r e G is not c o m p a c t and t r e a t p r o p e r actions. The Slice T h e o r e m w i l l be proved.

C o n v e n t i o n 1.1. We a l w a y s w o r k in the c a t e q o r y of c o m p a c t l y g e n e r a t e d s p a c e s (see S t e e n r o d [1967] or W h i t e h e a d [1978], p. 17 - 21). We r e c a l l that a c o m p a c t l y g e n e r a t e d space X is a H a u s d o r f f space such that a sub- set A c X is c l o s e d if and e n l y if its i n t e r s e c t i o n w i t h any c o m p a c t s u b s e t is closed, o

In t h i s s e c t i o n G is a t o p o l o q i c a l q r o u p (which is a s s u m e d to be com- p a c t l y g e n e r a t e d by c o n v e n t i o n 1.1). A t o p o l o q i c a l q r o u p w h i c h is a H a u s d o r f f space and l o c a l l y c o m p a c t is c o m p a c t l y g e n e r a t e d . E x a m p l e s are Lie g r o u p s (see B r e d o n [1972] I.I for the d e f i n i t i o n ) . We a l w a y s a s s u m e that a s u b g r o u p H c G is closed.

D e f i n i t i o n 1.2. Let (X,A) be a p a i r of G - s p a c e s such t h a t A / G is a H a u s - d o r f f space. A r e l a t i v e G - C W - c o m p l e x s t r u c t u r e on (X,A) c o n s i s t s of

a) a f i l t r a t i o n A = X _ ] c X O c X 1 c X 2 c . . . __0f x = U x n = - I n

a n d

b) a c o l l e c t i o n {e~ I i l £ In} of G - s u b s p a c e s e~ c X for e a c h n ~ O

- - l n

w i t h the p r o p e r t i e s :

i) X has the w e a k t o p o l o q y w i t h r e s p e c t to the f i l t r a t i o n {Xnln ~ -I}.

i.e., B c X is c l o s e d if and o n l y if B N X n c X n is c l o s e d for any n a -I,

ii) For e a c h n >- O t h e r e is a G - p u s h - o u t

± i i 6 I n

± ± i E I n

G/Hi sn~1 i E I n

x > Xn. I

[

n I

G/H i x D n > X n

such that e~ = On(G/Hi - - i X int D n) .

I f A i_~s empty we call X a G-CW-complex. o

The G-subspace X n is the n-skeleton of (X,A). The G-subspaces e i are n called the open cells. The number n is its dimension and the conjugacy class of subgroups (H i ) its type. The map q~ is an attaching map and the p~ir (Q~,q~) : G/H i x (Dn,S n-l) ~ (Xn,Xn_ I) a characteristic map

O n ~ n sn-1

for e~ We call ~n:= z" i i (G/H i x D n) a closed cell and De = q i ( G / H i x ) its boundary, we emphasize that the filtration and the open cells are part of the structure of a relative G - C W - c o m p l e x but not the attaching or characteristic maps. An isomorphism f : (X,A) ~ (Y,B) of relative G-CW-complexes is a G - h o m e o m o r p h i s m of pairs respecting the skeletal filtration and mapping open cells b i j e c t i v e l y to open cells.

Here is a list of basic facts proved later.

1.3. The open and closed ceils are already determined by the skeletal filtration. Namely, the open cells e~ are the G-components of X n \ X n _ I, i.e. the lifts of the components of (X n \ Xn_I)/G. The closed cell e~l is the closure of e~l in X and ~e~ = e~ \ e hi. In par- ticular e~ is open in X n and e~ i is closed in X. A subset C c X is closed if and only if C N A in A and C N ~n in e~ is closed o

i

1.4. Let H c G be normal and (X,A) a relative G-CW-complex such that A/H is a Hausdorff space. Then (X/H,A/H) has a canonical G/H-CW-struc-

a) H is c o m p a c t .

b) G x is c o m p a c t for e a c h x 6 X \ A , c) G / H is d i s c r e t e .

N a m e l y , t h e n - s k e l e t o n is X n / H and t h e o p e n n - c e l l s a r e

{ e ~ / H I i 6 In} , if X n is t h e n - s k e l e t o n and [e~ I i l 6 I n } t h e o p e n n - c e l l s of X. D

1.5. If (X,A) is a r e l a t i v e G - C W - c o m D l e x , t h e i n c l u s i o n A ~ X is a G-co- f i b r a t i o n . D

A G - s p a c e X is o b t a i n e d f r o m t h e G - s D a c e A b y a t t a c h i n q n - d i m e n s i o n a l c e l l s if t h e r e is a G - p u s h - o u t

II G / H i x S n-1 ... ~ A i 6 I

: f

± I G / H i x D n ~ X

i £ I

1.6. L e t A b e a ( c o m p a c t l y q e n e r a t e d ) G - s p a c e s u c h t h a t A / G is a H a u s - d o r f f space. T h e n A / G is a l s o c o m p a c t l y q e n e r a t e d ( S t e e n r o d [1967] 2.6).

L e t (X,A) b e a G - p a i r w i t h a G - f i l t r a t i o n {X n I n ~ -1}, X_1 = A, X = U x n s u c h t h a t X h a s t h e w e a k t o p o l o q y w i t h r e s p e c t t o t h i s f i l t r a t i o n a n d X n is o b t a i n e d f r o m X n _ I b y a t t a c h i n q n - d i m e n s i o n a l cells. T h e n X is c o m p a c t l y g e n e r a t e d and (X,A) a r e l a t i v e G - C W - c o m p l e x . o

E x a m p l e 1.7. L e t G be a f i n i t e q r o u p a n d X be a C W - c o m p l e x . S u p p o s e t h a t G a c t s c e l l p r e s e r v i n g on X, i.e. if e is an o p e n c e l l o f X , t h e n g e is a g a i n an o p e n c e l l a n d ge = e implies, t h a t lq : e ~ e x ~ q x is t h e identityj for all q E G. T h e n X is a G - C W - c o m p l e x ( c o m p a r e B r e d o n [1972]

I I I . 1 ) . m

t a t i o n . T h e u n i t s p h e r e S V h a s t h e f o l l o w i n g G ~ C W - c o m p l e x s t r u c t u r e . C h o o s e a b a s e { e l , . . . , e m } . L e t X be the c o n v e x h u l l o f

{ ± g e i I g 6 G, I ~ i s m}. Its b o u n d a r y ~X is G - h o m e o m o r p h i c to S V b y r a d i a l p r o j e c t i o n so t h a t it s u f f i c e s t o d e f i n e a G - C W - c o m p l e x s t r u c - t u r e o n ~X. N o w t h e r e is a s i m p l i c i a l c o m p l e x s t r u c t u r e o n ~X s u c h t h a t G a c t s s i m p l i c i a ! l y . C o n s i d e r its f i r s t b a r y c e n t r i c s u b d i v i s i o n . T h e n for a n y s i m p l e x o w i t h go = o m u l t i p l i c a t i o n w i t h g i n d u c e s t h e i d e n t i t y on it ( B r e d o n [1972] III 1.1). N O W a p p l y 1.7. o

1.9. L e t G b e a c o m p a c t L i e g r o u p and M a G - m a n i f o l d , i.e. M is a c o m - p a c t C ~ - m a n i f o l d a n d G a c t s b y a C ° - m a p G x M ~ M. T h e n M h a s t h e s t r u c - t u r e o f a G - C W - c o m p l e x . M o r e o v e r , a G - t r i a n q u ! a t i o n c a n b e c o n s t r u c t e d

(see I l l m a n [1983] p. 500). o

T o p r o v e t h e s t a t e m e n t s 1.3 t o 1.6 w e n e e d some m a t e r i a l a b o u t G - c o f i b - r a t i o n s a n d G - p u s h - o u t s .

A G - m a p i : A ~ X is c a l l e d a G - c o f i b r a t i o n if i(A) is c l o s e d in X a n d for a n y G - s p a c e Y, G - m a p s f : X ~ Y and h : A x I ~ Y w i t h f0i = h

o t h e r e is a G - m a p h : X x I ~ Y s a t i s f y i n g h o(i x i d ) = h and h o = f

A x O ~ i ½ X x O

Y

L e t (X,A) b e a G - p a i r . It is c a l l e d a G - N D R ( G - n e i g h b o u r h o o d d e f o r m a t i o n r e t r a c t ) if t h e r e are G - m a p s u : X ~ I a n d h : X x I ~ X s a t i s f y i n g :

i) A = u -I (O).

ii) h o = id.

iii) h t I A = id A for t 6 I.

iv) hi(x) 6 A for u(x) < I.

L e m m a 1.10.

a) If i : A ~ X is a G - c o f i b r a t i o n t h e n i is a c l o s e d e m b e d d i n q , i.e.

i(A) is c l o s e d in X a n d i : A ~ i(A) ~ G - h o m e o m o r p h i s m .

b) L e t (X,A) be a G - p a i r . T h e n i : A ~ X is a G - c o f i b r a t i o n if a n d o n l y i_~f (X,A) i_~s ~ G-NDR.

P r o o ~ a n a l o g o u s to t h e n o n - e q u i v a r i a n t c a s e (Strjm [1966], S t e e n r o d [1967], 7.1.)

A s q u a r e

f

A ~,Y

F

X ; Z

o f G - s p a c e s is a G - p u s h - o u t if f o r e a c h p a i r o f G - m a p s f' : y ~ U a n d j' : X ~ U w i t h f'f = j'j t h e r e is a G - m a p u : Z ~ U u n i q u e l y d e t e r m i n e d b y u J = f' a n d u F = j'

L e m m a 1.11. L e t j : A ~ X b e a G - c o f i b r a t i o n . T h e n a G - p u s h - o u t Z is

@ i v e n b y t h e a d j u n c t i o n s p a c e Y Uf X w h i c h is c o m p a c t l y q e n e r a t e d . J i_~s a G - c o f i b r a t i o n .

P r o o f : F o r G = I see S t e e n r o d [1967] 8.5. m

L e t X be a G - s p a c e a n d H c G a (closed) s u b q r o u p . F o r x 6 X let its iso- t r o p y g r o u p G x b e {q £ G I q x = x}. It is c l o s e d in G. L e t t h e H - f i x e d p o i n t set X H b e {x 6 X I G x m H}, X >H b e {x 6 X I G x = H, G x ~ H} a n d

X H be {x E X I G x = H}. T h e n X H and X >H are c l o s e d in X and x H \ x > H = X H.

If H and K are s u b g r o u p s we call H s u b c o n j u q a t e d to K if gHg -I c K h o l d s for s u i t a b l e g and w r i t e (H) c (K). D e f i n e X (H) = Ix 6 X ! (G x) m (H)}, x>(H) = (X 6 X I (G x) = (H) r (G x) • (H)} and ×(H) = {x 6 X I (G x) = (H)}.

T h e n X (H) _, X >(H) and X(H ) are G - s u b s ~ a c e s and x ( H ) \ x >(H) _ = X ( H ) •

For H c G let NH be its n o r m a l i z e r {g 6 G I ~H~ -I = H} and WH = N H / H its W e y l group. T h e G - a c t i o n on X i n d u c e s W H - a c t i o n s on X H, X >H and X H.

L e m m a 1.12. Let i : A ~ X be a G - c o f i b r a t i o n . a) Its r e s t r i c t i o n to H c G is a H - c o f i b r a t i o n .

b) For H c G the m a p i H : A H ~ X H is a W H - c o f i b r a t i o n .

c) The m a p i/H : A / H ~ X/H is a G / H - c o f i b r a t i o n i f A / H and X/H are H a u s d o r f f s p a c e s and H c G n o r m a l .

Proof: Use L e m m a 1.10. For G - N D R - s the s t a t e m e n t s are o b v i o u s , The as- s u m p t i o n a b o u t A/H and X/H in c) q u a r a n t e e s t h a t t h e y are c o m p a c t l y ge- n e r a t e d ( S t e e n r o d [1967] 2.6). []

L e m m a 1.13. C o n s i d e r t h e G - p u s h - o u t

f A

F X

Y

A s s u m e that j is a G - c o f i b r a t i o n .

a) T h e r e s t r i c t i o n t__o K c G is a K - p u s h - o u t .

b) T a k i n @ the H - f i x e d p o i n t set Y i e l d s ~ W H - p u s h - o u t .

c) Let H c G be n o r m a l and A/H, X / H and Y / H be H a u s d o r f f spaces. T h e n w__ee get a G / H - p u s h - o u t by d i v i d i n g out the H - a c t i o n .

Proof. We o n l y v e r i f y c). We m u s t show t h a t F/H + J/H : X/H + Y/H ~ Z/H is an i d e n t i f i c a t i o n . T h i s f o l l o w s from the fact that for a H - m a p

u : A ~ B , w h i c h is an i d e n t i f i c a t i o n , a l s o u / H : A / H ~ B / H is an i d e n t i f i c a t i o n .

A ~ B

prAi IPrB

A/H ~ B/H

u/H

N a m e l y , p r A and u / H o pr A = p r B o u are i d e n t i f i c a t i o n s . T h e m a p j/H is a G / H - c o f i b r a t i o n b y L e m m a 1.12. c). s

N o w w e c o m e to t h e p r o o f of 1.3. to 1.6. R e c a l l t h a t X n is t h e G - p u s h - o u t

It qi

it G / H i _× S n-1 _~ X n _ 1

i t G / H i × D n ~ X n

and j is a G - c o f i b r a t i o n . H e n c e Xn_1 ~ X n is a G - c o f i b r a t i o n . N o w 1.5 f o l l o w s f r o m t h e e q u i v a r i a n t v e r s i o n o f W h i t e h e a d [1978] I. 6.3. In 1.6.

the o n l y p r o b l e m is t o s h o w t h a t X is c o m p a c t l y g e n e r a t e d . O n e s h o w s in- d u c t i v e l y u s i n g L e m m a 1.11. t h a t e a c h X n is c o m p a c t l y g e n e r a t e d a n d a p - p l i e s W h i t e h e a d [1978] I. 6.3.

B y L e m m a 1.12. a n d 1.13. w e h a v e the G / H - p u s h - o u t w i t h j/H a c o f i b r a t i o n

II qi/H

(G/H i )/H x S n~1 Xn~ I/H

xt Oi/H

]I (G/Hi)/H x D n Xn/H

The conditions appearing in 1.4. g u a r a n t e e that HH i is closed in G (tom Dieck [1987], 1.3.1.). Since H is normal, HH i is a closed subgroup in G so that (G/Hi)/H is G / H - h o m e o m o r p h i c to the c o m p a c t l y g e n e r a t e d G/H- space (G/H)/(HHi/H). Now 1.4. follows using W h i t e h e a d [1978] 1.6.4. to verify that X/H has the weak t o p o l o g y w i t h respect to {Xn/H I n ~ -I}.

For 1.3. c o n s i d e r the p u s h - o u t above for H = G 31 qi/G

/I S n Xn_ I/G

/I D n Xn/G

As above one shows that Xn/G is a H a u s d o r f f space so that Qi/G(Dn) C X n / G is compact and in p a r t i c u l a r closed. T h e n ~ni = Qi(G/Hi x D n) c X n is closed in X n. Now it is easy to prove 1.3.

E x a m p l e 1.14. Let G be the m u l t i p l i c a t i v e group of p o s i t i v e real num- bers. Define an action

p : G x ~ ~ ~ (g,r) ~ gr

Then ~ / G is not a H a u s d o r f f space. Let q : G x S ° ~ ~ be the G-map in- duced from the inclusion S ° c ~. C o n s i d e r the G - p u s h - o u t

q

G x S ° ) ]R

G x D I Q 3 X J

T h e n Q ( G x D I) c X is o p e n b u t n o t c l o s e d in X s i n c e image q is o p e n but not c l o s e d (compare 1.3.). []

R e m a r k 1.15. In the d e f i n i t i o n of a G - C W - c o m p l e x g i v e n f,e, in I l l m a n [1974] 1.2. it is p a r t of the d e f i n i t i o n of a G - C W - c o m p l e x t h a t e i is -n

c l o s e d in X. H e n c e o u r d e f i n i t i o n r e q u i r e s less. But we h a v e p r o v e n by 1.3. t h a t b o t h d e f i n i t i o n s agree, H o w e v e r , we h a v e g a i n e d t h a t in 1.6.

a t t a c h a r b i t r a r i l y c e l l s a n d h a v e n o t to care w h e t h e r Q ( G / H i x D n) we can

is a l w a y s closed. T h i s is v e r y p l e a s a n t w h e n we w a n t to m a k e a G - m a p h i g h l y c o n n e c t e d by a t t a c h i n g cells, a

N o w w e n e e d some b a s i c facts a b o u t p r o p e r m a p s and p r o p e r a c t i o n s .

A m a p f : X ~ Y is p r o p e r if f is c l o s e d and f-1(y) is c o m p a c t for any y £ Y. A g e n e r a l r e f e r e n c e for p r o p e r m a p s is B o u r b a k i [1961] I.IO.. A m a p f : X ~ Y b e t w e e n c o m p a c t l y g e n e r a t e d s p a c e s is p r o p e r if and on-

ly if f-1(C) is c o m p a c t for any c o m p a c t C c Y (use B o u r b a k i [1961]

I.IO.2. p r o p o s i t i o n 6).

L e m m a 1.16.

a) C o n s i d e r m a p s f : X ~ Y and g : Y ~ Z.

i) I_~f f and g are p r o p e r , t h e n gf i_~s p r o p e r . ii) I f gf is p r o p e r , f i_~s proper.

iii~ I_~f gf i__ss p r o p e r and f s u r j e c t i v e , t h e n g i__ss p r o p e r .

b) I_~f f : X ~ Y i__ss p r o p e r and B c Y t h e n the i n d u c e d m a p f-1 (B) ~ B i_~s proper. I_ff f is p r o p e r and A c X is c l o s e d f IA : A ~ Y i__ss p r o - per.

c) If f : X ~ Y a n d f' : X' ~ Y' a r e p r o p e r t h e n f x f' : X x X' y x y' is p r o p e r . If f : X ~ Y a n d g : X ~ Z a r e p r o p e r , t h e n f x g : X ~ y x Z is p r o p e r .

d) A p r o j e c t i o n p r : X x y ~ Y i__ss p r o p e r if a n d o n l y i_~f X i__ss c o m p a c t . e) C o n s i d e r a p u s h - o u t

f

A % X

f f

j J

F

Y ~ g

I_~f j is a c l o s e d e m b e d d i n g a n d f i_~s p r o p e r t h e n F i s p r o p e r . f) C o n s i d e r a p u l l - b a c k

F

Z ) X

If f is p r o p e r , t h e n F is p r o p e r .

g) L e t {Xn J n ~ -I] resp. {Yn I n ~ -1} b e a c l o s e d f i l t r a t i o n f o r X resp. Y s u c h t h a t X resp. Y h a s the w e a k t o p o l o g y . L e t f : X -- Y b e a m a p s u c h t h a t f ( X n \ X n _ I) c y n \ Y n _ l h o l d s for n ~ -I. A s s u m e t h a t e a c h m a p fn : X n ~ Yn i_~s p r o p e r . T h e n f i__ss p r o p e r .

Proof: T h e v e r i f i c a t i o n of a), b), c) a n d d) g i v e n in B o u r b a k i [1961]

I. I0 is e a s i l y c a r r i e d o v e r t o c o m p a c t l y g e n e r a t e d s p a c e s .

e) G i v e n C c y t h e s u b s e t F(C) c Z is c l o s e d if a n d o n l y if C U f - l f ( c N A) c y is c l o s e d and f(C N A) c X is c l o s e d . f) C o n s i d e r t h e c o m m u t a t i v e d i a g r a m

F x P

i d x f

X x Y ~ X x A

F

7, ~ X

T h e m a p id x f is p r o p e r b y a s s u m p t i o n a n d c). F r o m t h e e x p l i c i t c o n - s t r u c t i o n o f a m o d e l f o r Z as t h e p r e i m a g e o f t h e d i a g o n a l u n d e r f x p : X x y ~ A x A w e d e r i v e t h a t F x p is a c l o s e d e m b e d d i n g a n d h e n c e p r o p e r . N o w a p p l y a).

g) is l e f t to t h e r e a d e r . D

D e f i n i t i o n 1.17. A G - s p a c e X is p r o p e r if t h e m a p

is p r o p e r . []

0 x : G x X -- X x X (g,x) ~ (x,gx)

L e m m a 1.18. If G is c o m p a c t p a n y G - s p a c e is p r o p e r .

P r o o f : T h e p r o j e c t i o n p r : G x X ~ X is p r o p e r ( L e m m a 1.16 d). C o m - p o s i n g it w i t h t h e G - h o m e o m o r p h i s m G x X ~ G x X (g,x) ~ (g,gx)~ d e - f i n e s a p r o p e r m a p p : G x X ~ X b y L e m m a 1.16 a. T h e n 8 x = p r x p is p r o p e r b y L e m m a 1.16 c. o

L e m m a 1.19. L e t X b e a p r o p e r G - s p a c e . T h e n X / G b e l o n g s a l s o t o t h e c a t e g o r y o_~f c o m p a c t l y g e n e r a t e d s p a c e s . W e h a v e f o r x 6 X:

i) G ~ X g ~ g x i_~s p r o p e r . ii) G x i_ss c o m p a c t .

iii) T h e m a p G / G x ~ G x g G x ~ gx is a G - h o m e o m o r p h i s m . iv) T h e o r b i t G x is c l o s e d in X.

P r o o f : B o u r b a k i [1961] III. 4.2. p r o p o s i t i o n 3 + 4.

L e m m a 1.19 s h o w s t h a t t h e G - s p a c e ~ of Example 1.14 is n o t p r o p e r .

L e m m a 1.20. L e t X be a p r o p e r G - s p a c e and C ~ c o m p a c t s p a c e w i t h t r i v i a l G - a c t i o n . T h e n any G - m a p G / H x C ~ X i_~s p r o p e r . I__nn p a r t i c u l a r its i m a g e is c l o s e d in X.

Proof: The s u b g r o u p H is c o m p a c t by L e m m a 1.19. so t h a t G ~ G/H is p r o p e r (Bourbaki [1961] III.4.1. cor° 2). H e n c e we can a s s u m e H = 1 by L e m m a 1.16. a and c.

L e t D c X be the c o m p a c t s u b s e t f({e} x C) and f : C ~ D be the i n d u c e d map. By L e m m a 1.16. the m a p s id x f : G x C ~ G x D , v : G x D ~ D x X

(g,x) ~ (x,gx) and pr : D x X ~ X and h e n c e t h e i r c o m p o s i t i o n f are proper. D

L e m m a I .21. C o n s i d e r t h e G - p u s h - o u t s u c h t h a t Y and X i are p r o p e r G- spaces, fi i_~s p r o p e r and Ji a G - c o f i b r a t i 0 n ,

II f. l i 6 I

li A i ) Y

i £ I I

I II Ji J

i 6 I

II X i ) Z

i 6 I F

T h e n Z is a p r o p e r G - s p a c e .

Proof: If C c Z is c o m p a c t C D F ( X i ~ A i) • @ h o l d s o n l y for f i n i t e l y m a n y i 6 I as (Xi,A i) is a G - N D R - p a i r . H e n c e we c a n a s s u m e t h a t I is

finite. W r i t e X = i 6 I

± I X i. C o n s i d e r the d i a g r a m e z

G x Z .... ~2 Z x Z

]'

^I

0Xit 8y

G x (XitY) = G x X i t G x y % X x X i t Y x Y

By L e m m a I. 11. J is a G - c o f i b r a t i o n and h e n c e a c l o s e d e m b e d d i n g by L e m m a 1.10. T h e n J x J is p r o p e r (Lemma 1.16. c). T h e m a p F x F is p r o p e r

by L e m m a I. 16. c and e. S i n c e X and Y are p r o p e r G - s p a c e s by a s s u m p t i o n e Z o id x (F/.iJ) = (F x F /.t J x j) o (SXll~y) is p r o p e r (Lem/na 1.16. a).

T h e n 8 z is p r o p e r (Lemma 1.16. a). o

L e m m a 1.22. L e t {Xn I n ~ -I} b__ee a c l o s e d f i l t r a t i o n of X such t h a t X has t h e w e a k t o p o l o g y . I f e a c h X i s p r o p e r , t h e n X i s p r o p e r .

n ~ ~

Proof: Use L e m m a 1.16. g and S t e e n r o d [1967] 10.3. o

T h e o r e m 1.23. L e t (X,A) be a r e l a t i v e G - C W - c o m p l e x . T h e n X is a p r o p e r G - s p a c e if a n d o n l y i f A i__ss p r o p e r and G x c o m p a c t for all x 6 X ~ A. I__nn p a r t i c u l a r ~ G - C W - c o m p l e x X i~s p r o p e r if and o n l y i f G x i__ss c o m p a c t for e a c h x 6 X.

Proof: One s h o w s i n d u c t i v e l y t h a t any X n is p r o p e r . If H is c o m p a c t , G / H is a p r o p e r G - s p a c e . By L e m m a 1.16. the G - s p a c e G / H x D n is p r o p e r . N o w the i n d u c t i o n step f o l l o w s f r o m L e m m a 1.20. and 1.21. F i n a l l y a p p l y L e m m a 1.22. to s h o w t h a t X is p r o p e r , o

1.24. L e t X be a free G - C W - c o m p l e x . T h e n X is p r o p e r by T h e o r e m 1.23.

S i n c e X is f r e e , t h i s is e q u i v a l e n t to i m a g e ~ S x ) C X x X b e i n g c l o s e d and the m a p i m a g e C S x ) ~ G (x,gx) ~ g b e i n g c o n t i n u o u s (tom D i e c k [1987]

1.3.20). We w i l l s h o w in T h e o r e m 1.37. t h a t X is l o c a l l y t r i v i a l so t h a t X ~ X / G is a p r i n c i p a l G - b u n d l e (see H u s e m o l l e r [1966], 4.2.2). By 1.4. X / G is a C W - c o m p l e x . o

1.25. T h i s p r o c e s s c a n be r e s e r v e d . Let p : E ~ B be a p r i n c i p a l G- b u n d l e and A c B a s u b s p a c e . T h e n a r e l a t i v e C W - c o m p l e x s t r u c t u r e on

(B,A) l i f t s to a G - C W - c o m p l e x s t r u c t u r e on (E,p-I(A)). N a m e l y , let E n be

-1 n

p (Bn). If {ei I i E I n } are the o p e n n - c e l l s of (B,A), let {p-1(e~) I i E In} be the o p e n n - c e l l s of (E,p-I(A)). S i n c e B h a s the w e a k t o p o l o - gy w i t h r e s p e c t to {Bn I n ~ -I} the same is true for E and {En J n ~ -I]

(see W h i t e h e a d h978] , X I I I . 4 . 1 ) .

N o w B n is the G - p u s h - o u t

it S n-1 q

• ~, B n _ I i 6 I

n

f

Q i t D n

i 6 I n n

By the L e m m a I .26. b e l o w we o b t a i n a G - p u s h - o u t by the p u l l - b a c k con- s t r u c t i o n a p p l i e d to P n : E n B n

q E n ) En_ I

Q E ~ E

n ~ n

= J E n

S i n c e D n is c o n t r a c t i b l e , t h e r e is a G - h o m e o m o r p h i s m of G - p a i r s (Q En, q E n) ~ it G x (Dn,S n-l) ( H u s e m o i l e r [1966], 5 . 1 0 . 3 . ) . Q

i £ I n

L e m m a 1.26. C o n s i d e r the p u s h - o u t w i t h j a c o f i b r a t i o n

f

A , , ~ Y

x ,,,~ z

F

L e t p : E ~ Z be a f i b r a t i o n . T h e n the p u l l - b a c k c o n s t r u c t i o n [ i e l d s p u s h - o u t w i t h ~ a c o f i b r a t i o n .

• • ~ ,

f J E ) J E

F E -~ E

Proof: The m a p j is a c o f i b r a t i o n by W h i t e h e a d [1978] 1.7.14. It r e m a i n s to s h o w t h a t J U~ F : J E Uf~j~ E F E ~ E is an i d e n t i f i c a t i o n . By a s s u m p t i o n J Uf F : Y Uf X ~ Z is an i d e n t i f i c a t i o n . By S t e e n r o d

[1967] 4.4. id x (j Uf F) and h e n c e

(id x J) U(id x f) (id x F) : E x B I Uid x f E x B 2 ~ E x B is an i d e n t i f i - cation. R e s t r i c t i n g it to {(e,p(e)) I e 6 E} c E x B y i e l d s just J U~ F. D

1.27. If (X,A) is a r e l a t i v e G - C W - c o m p l e x and (Y,B) a r e l a t i v e G ' - C W - c o m p l e x t h e n (X,A) x (Y,B) has a r e l a t i v e G x G ' - C W - c o m p l e x s t r u c t u r e . The k - s k e l e t o n (X x Y ) k is U x n x Ym" T h e n {(X x Y)k J k ~ -I} is a

n + m = k

c l o s e d f i l t r a t i o n of X x y s u c h t h a t X x Y has t h e weak t o p o l o g y ( S t e e n r o d [1967] 10.3.).

If {e~ J i E I n ] are the o p e n n - c e l l s of X and { j I j E Jm ] the open m-cells fm of Y t h e n {e; x f~ I i E I n j £ J m ' n + m = k ) are the o p e n k - c e l l s of X x Y.

3

T h e c h a r a c t e r i s t i c m a p for e; x fm is the p r o d u c t of t h e o n e s for e~

3 z

and f ; , i f w e i d e n t i f y (G/H i x D n) x (G'/H~ x D m) and (G x G ' ) / ( H i x Hi) x D n+m. D

1.28. L e t (X,A) be a r e l a t i v e G - C W - c o m p l e x . A r e l a t i v e G - C W - s u b c o m p l e x (Y,B) is a p a i r s a t i s f y i n g

i) Y is a G - s u b s p a c e of X.

ii) B is a c l o s e d s u b s p a c e of A.

iii) Y is the u n i o n of A and a c o l l e c t i o n of o p e n c e l l s w h o s e b o u n d a r i e s a l s o b e l o n g t o Y.

T h e n (Y,B) i t s e l f is a r e l a t i v e G - C W - c o m p l e x w i t h Yn = Xn D Y and Y is

c l o s e d in X. If A a n d B a r e e m p t y , ( X , Y ) is c a l l e d a ~ a i r o f G - C W - c o m - p l e x e s , a

1.29. C o n s i d e r t h e G - p u s h - o u t

f

A % Y

X ~ Z

L e t (X,A) b e a r e l a t i v e G - C W - c o m p l e x . T h e n t h e r e is a r e l a t i v e G - C W - c o m p l e x s t r u c t u r e o n (Z,Y) s u c h t h a t t h e r e l a t i v e h o m e o m o r p h i s m (F,f) m a p s X n t o Z n a n d o p e n c e l l s b i j e c t i v e l y t o o p e n c e l l s .

A s s u m e t h a t (X,A) is a p a i r of G - C W - c o m p l e x e s a n d f is c e l l u l a r i.e.

f(An) c Y n for all n ~ O. T h e n w e g e t t h e s t r u c t u r e o f a p a i r o f G - C W - c o m p l e x e s o n (Z,Y).

In p a r t i c u l a r X / A is a G ~ C W - c o m p l e x if (X,A) is a p a i r of G - C W - c o m p l e x e s . G i v e n a c e l l u l a r G - m a p f : X ~ Y b e t w e e n G - C W - c o m p l e x e s its m a p p i n g c y l i n d e r a n d m a p p i n g c o n e g e t G - C W - s t r u c t u r e s b y 1.27. a n d t h e c o n s t r u c - t i o n a b o v e , s

1.30. C o n s i d e r t h e s p a c e m a p ( X , Y ) o f m a p s X ~ Y t o p o l o g i z e d as in S t e e n r o d [1967] §5 . If X a n d Y are G - s p a c e s , G a c t s o n m a p ( X , Y ) b y f ~ 1 o f o ig_l w h e r e 1 is m u l t i p l i c a t i o n w i t h G. T h e G - f i x e d p o i n t

g g

s e t m a p ( X , Y ) G c m a p ( x , y ) c o n s i s t s o f a l l G - m a p s f : X ~ Y. C o n s i d e r t h e m a p s

a n d

: m a p ( G / H , X ) G ~ X H ~ ~ ~(eH)

: X H ~ m a p ( G / H , X ) G x (@(x) : gH ~ gx)

U s e S t e e n r o d [1967] 5.2. a n d 5.8. t o s h o w t h a t t h e y are c o n t i n u o u s . H e n c e a n d ~ a r e i n v e r s e h o m e o m o r p h i s m s , a

we n e e d some f a c t s a b o u t h o m o g e n o u s spaces.

L e m m a 1.31. Let H and K be s u b g r o u p s of G.

a) T h e r e is an e q u i v a r i a n t m a p G / H ~ G / K if and o n l y i f (H) = (K) holds.

b) I_~f g £ G an d g - I H g c K, t h e n we ~ g t a w e l l d e f i n e d G ~ m a P

R : G / H ~ G/K g ' H ~ g ' g K g

c) E v e r y G - m a p G / H ~ G / K is of the f o r m R . We h a v e Rg g o n l y i f g-lg, 6 K holds.

= Rg, if and

d) A s s u m e that G i__ss c o m p a c t or that G is a L i e g r o u p and H c G compact.

T h e n we h a v e g - I H g c H ~ g - I H g = H for any g £ G and o b t a i n a h o m e o - m o r p h i s m o f t o p o l o g i c a l g r o u p s

WH ~ m a p (G/H,G/H) G gH ~ Rg_1

Proof: a), b) a n d c) are v e r i f i e d in t o m D i e c k [1987] I.I.14. T h e p r o o f of d) for c o m p a c t G can be f o u n d in B r e d o n [1972] O.1.9. L e t G be a L i e g r o u p and H compact. S u p p o s e g - I H g c H for g 6 G. T h e n g - I H g is a sub- m a n i f o l d of H. T h i s i m p l i e s g - l H g = H b e c a u s e for a c o n n e c t e d s u b m a n i - fold M of a c o n n e c t e d m a n i f o l d N w i t h d i m M = d i m M a l r e a d y M = N h o l d s and H has f i n i t e l y m a n y c o m p o n e n t s . F i n a l l y a p p l y 1.30.

E x a m p l e 1.32. W e w a n t to i l l u s t r a t e by t h i s e x a m p l e that t h e c o n d i t i o n s in L e m m a 1.31. d) are n e c e s s a r y .

Let G c GL(2, ~) be the Lie g r o u p of m a t r i c e s o v e r ~ of the s h a p e

a

0 a -I a,b 6 3~, a • O

D e n o t e by H c G t h e s u b g r o u p of all m a t r i c e s A w i t h a = I and b 6 Z~.

O n e e a s i l y c h e c k s

-I 2 (a b > <1 n> Ia b > /I a nh

' 0 a - 1

0

0

\0

Hence AHA -I c H is e q u i v a l e n t to a 2 £ ~ w h e r e a s AHA -I = H is e q u i v a l e n t to a = ±I. This gives a c o u n t e r e x a m p l e w i t h G a Lie group.

^

If we s u b s t i t u t e ~ by the p-adic rationals ~p and ~ by the p - a d i c num- bers Z p we o b t a i n a c o u n t e r e x a m p l e where H is compact.

The next result is one of the m a i n p r o p e r t i e s of Lie groups.

T h e o r e m 1.33. Let G be a Lie g r o u p and H and K b_~e subgroups. Suppose that H i_~s compact.

Then G/K H i__ss th__~e d i s j o i n t union of its W H - o r b i t s or, equivalently, (G/KH)/wH is discrete. If G i__ss compact, G/K H is the d i s j o i n t union of finitely m a n y WH-orbits.

Proof: If G is compact this is shown in Bredon [1972] II.5.7. By in- specting the proof we see that it works also for G a Lie group and com- pact H if the result in Bredon [1972] II.5.6. is still true. But this is v e r i f i e d for G a Lie group and H compact in M o n t g o m e r y - Z i p p i n [1955], p. 216. D

1.34. Let H be a subgroup of G. If (X,A) is a relative H - C W - c o m p l e x , t h e n (ind X, ind A) = G x H (X,A) has a c a n o n i c a l relative G - C W - c o m p l e x struc- ture. This follows from the i d e n t i t y G x H H/K = G/K. o

1.35. Let G be a Lie group and H a subgroup w i t h dim H = dim G. Consider the r e s t r i c t i o n res G/K of the h o m o q e n o u s G-space G/K. Since G/H is dis- crete (res G/K)/H is discrete. Hence res G/K is a d i s j o i n t u n i o n of ho- m o g e n o u s H - s p a c e s and has t h e r e f o r e a canonical H-CW-structure. This

carries over to G-CW-complexes. If (X,A) is a r e l a t i v e G - C W - c o m p l e x then res (X,A) has a canonical relative H-CW-structure.

T h e a s s u m p t i o n d i m H = d i m G is e s s e n t i a l . F o r e x a m p l e , a h o m o g e n o u s G - s p a c e G / K h a s e x a c t l y o n e G - C W - s t r u c t u r e . If d i m G Z I it is n o t o b v i o u s t h a t G / K j u s t as a s p a c e h a s a C W - c o m p l e x s t r u c t u r e a n d t h a t t h e r e is e v e n a c a n o n i c a l one. W e w i l l h a v e t o d e a l at s e v e r a l p l a c e s w i t h t h e p r o b l e m t h a t t h e r e s t r i c t i o n o f a G - C W - c o m p l e x t o a s u b g r o u p H w i t h d i m H < d i m G h a s n o o b v i o u s H - C W - s t r u c t u r e . o

1.36. L e t G b e a L i e g r o u p a n d H c G c o m p a c t . F o r e a c h K c G w e h a v e a c a n o n i c a l W H - C W - s t r u c t u r e o n G / K H b y Theorem 1.33. Henoe for a n y r e l a t i v e G - C W - c o m p l e x (X,A) t h e r e is a c a n o n i c a l r e l a t i v e W H - C W - s t r u c t u r e on

(X H A M) o

t

W e m a k e s o m e r e m a r k s a b o u t s l i c e s . L e t G b e a t o p o l o g i c a l g r o u p a n d X a G - s p a c e . A s l i c e S at x 6 X is a G x - S U b s p a c e S c X s u c h t h a t GS is an o p e n n e i g h b o u r h o o d f o r x a n d ~ : G x G S ~ GS a G - h o m e o m o r p h i s m . T h e n

x GS is c a l l e d a t u b e a r o u n d t h e o r b i t Gx.

T h e o r e m 1.37. S l i c e T h e o r e m f o r G - C W - c o m p l e x e s .

L e t G be a t o p o l o g i c a l g r o u p a n d (X,A) a r e l a t i v e G - C W - c o m p l e x . A s s u m e t h a t A is p r o p e r , t h e r e is a s l i c e a 6 S c A i__nn A f o r a n y a 6 A a n d G x i__ss c o m p a c t f o r e a c h x 6 X ~ A . T h e n t h e r e is a s l i c e S at x in X f o r e a c h x 6 X.

P r o o f : L e t x 6 X b e g i v e n . C h o o s e n ~ -I s u c h t h a t x l i e s in X n b u t n o t in X n _ I. W e c o n s t r u c t i n d u c t i v e l y f o r m = n, n + 1,... G x - S U b S e t s S m c X m s u c h t h a t S m + I D X m = Sm, GS m is o p e n in X m a n d ~ m : G X G S m ~ G S m '

x

g , y ~ g y is a G - h o m e o m o r p h i s m . N o t i c e f o r t h e s e q u e l t h a t X is a p r o p e r G - s p a c e b y T h e o r e m 1.23.

T h e i n d u c t i o n b e g i n m = n f o l l o w s for n = -I f r o m t h e a s s u m p t i o n a b o u t A. If n Z O h o l d s t h e r e is an o p e n c e l l e n c o n t a i n i n g x. S i n c e e n is o p e n in X n a n d G - h o m e o m o r p h i c t o G / G x x i n t D it s u f f i c e s t o f i n d a s l i c e a r o u n d a n y p o i n t (gGx,Y) 6 G / G x x i n t D. B u t f o r a n y o p e n n e i g h b o u r h o o d

U of y in int D the G x - S e t gG x x U is a slice.

We come to the i n d u c t i o n step from m - I to m for m ~ I ~ n° C o n s i d e r the G - p u s h - o u t

11 i 6 I

/I i E I

ii qi i 6 I

x S m-1

G/Hi ) Xm- I

i £ I

G / H i x D m ~, X m

Let U i c G/H i x S m-1 be qi1(Sm_1 ). Define V i c G/H.I x D m as

{(gHi,tu) J (g~,u) 6 U i, I/2 < t S I}. N o t i c e that U i and V i are G x - S U b - sets and V i is G x - h o m e o m o r p h i c to U i x ]I/2,1]. Define S m as the u n i o n Sm_ I U iU6 1 Qi(Vi). we have by c o n s t r u c t i o n Sm N Xm_ I = Sm_ I.

By a s s u m p t i o n ~m-1 : G x G Sm_ I ~ GSm_ 1

x

l o w i n g d i a g r a m c o m m u t e s

is a G - h o m e o m o r p h i s m . The fol-

~U i

G x G U i > GU

X

id x G qiJUi

I

G x G qi(Ui) > qi(GUi)

x ~ m _ 1 1 G x G qi(Ui)

x

q i J G U i

The m a p ~Ui is b i j e c t i v e and c o n t i n u o u s , ~ m _ I T G x G qi(Ui) a G - h o m e o m o r - x

p h i s m and the v e r t i c a l m a p s are p r o p e r by L e m m a t a 1.16. and 1.20. since