Equivariant Z-Theory I

Math. Z. 203.503-526 (1990)

Mathematische zeitschrift

,(', Springer-Verlag 1990

Equivariant Z-Theory I

Wolfgang Lück 1' * and Ib Madsen 2

1 Mathematisches I n s t i t u t , G e o r g - A u g u s t U n i v e r s i t ä t , D - 3 4 0 0 G ö t t i n g e n , F c d e r a l R e p u b l i c o f G e r m a n y

2 Matematisk Institut. Aarhus Universitet, DK-8000 Aarhus C, Denmark

Introduction

This paper gives a new definition of G-equivariant surgery groups with better formal properties than previous definitions, and it calculates the equivariant groups in terms of usual L-groups in special cases.

The main new feature in our presentation is the systematic use of groupoids with.G-actions, and the concept of fibre transport (of G-bundles). This makes possi'ble a definition of equivariant L-groups much in the spirit of C.T.C. Wall's chapter g, l2l). The surgery groups are- bordism classes of degree one G-maps f : M - - + l 1 l c o v e r e d b y a G - b u n d l e m a p .f:TM @Ru--( together with a certain map from lü into a reference (space) R, which, very roughly, captures the first Stiefel-Whitney classes of ( and TN.

The normal map U,h defines a triple (no l'{, w,({), wt(lü)) as follows' The first term is the collection of fundamental groupoids zN[ of the various fixed sets, considered as a functor on the orbit category. The next two terms are t h e ' h o m o t o p y c l a s s e s ' o f t h e m a P S

w r ( 0 , w , ( l { ) : n G N - - - r l r G B ( G , n + k ) : B n + k

w h e r e B ( G , n * k ) c l a s s i f i e s ( l o c a l l y li n e a r ) G _ R n + f t b u n d l e s . W e r e m a r k t h a t w , ( 0 c a n b e d i f f e r e n t fr o m w , ( l { ) . H o w e v e r , t h e y d o a g r e e w h e n p u s h e d in t o t s 4 * k:nG BF(G, n+k) where BF(G, n+k) is the classifying s p a c e fo r ( l o c a l l y linear) G - S'*k fibrations.

One should expect the equivariant L-groups to depend on the above triple, generalizing Wall's L!,(nrN, w,). However, this is not quite enough. As pointed out in U; I9f, the definition from l2I, Chap.9] contains a bug: conjugation w i t h a n e l e m e n t g e r t N w i t h w r ( g ) : - 1 i n d u c e s th e i d e n t i t y a f t e r W a l l ' s g e o - metric definition, but it should be multiplication with - 1 after the algebraic definition of L-groups. The difficulty is one of base points and can be done

I I

* W.L. was supported by DFG

504 W . L ü c k a n d l . M a d s c n

away with by systematically working with orientation covers, or by passing to transports as we have preferred.

Given a G-lR'*ft bundle (or G-fibration) ( over lt' and a (homotopy class o f a ) p a t h o from r to y, there is the corresponding f i b r e t r a n s p o r t o*t (,-(.r.

This is a well-defined isomorphism class. The transport information can be c o l l e c t e d i n t o a n 6(G)-functor

tpr: nG lü * IBn + r.

T h e ' h o m o t o p y c l a s s ' o f t p * i s w , ( 0 . The tangent bundle Il{ gives another t r a n s p o r t

tpr: nG nu -- IBn ru , Bn+r.

Passing to the fibrewise one point compactifications one gets transports tp.:

and tp'" into tsd, * *. They are 'homotopic'

by the equivariant Freudentol rurp.n- s i o n t h e o r e m , because ( ( @v)' -(Tl/@Rft @v), for a suitable representation v . ln equivalent category terms, there exists an (,(G\-transformation

rp: tp'r-- )k tpiy.

There is no apriori choice for E, but given one, we have the 4-tuple R(.f, f, E): (n j

N, tps , tpN , rp),

which record sufficient information to define L-groups. Note that q..f identifies the tangent fibres of M and lü. This allows the definition of equivariant degree Deg(./'; E,.i), ancl in particular the concept of degree one.

Extracting t!. relevant properties of R ( f, f, d one obtains the concept of a (geometric) reference R and one can consider normal maps (/,i,d togäther with maps p: R (f, i, d --+ R. Bordism classes of such define the equivari int L- g r o u p s .y,(R):,9,(G; R), corresponding t o t h e 1 1 - g r o u p s i n [21, chap.9]. The use of reference R, independent of the range manifold lf, nur the usual advan- tages, namely one gets abelian groups with good functorial properties.

We prove the analog of Wall's Lt : Ü theorem, nu-.ly that every class i n 1 ( R ) h a s a r e p r e s e n t a t i v e f o r w h i c h p i s a n ' i s o m o r p h i s m ' ; th i s i s t h e n- Tt l e m m a .

O u r d e f i n i t i o n of 9!(R) makes sense i n t h e s m o o t h and in the locally linear l!' and_,Toy' categories. However, the TT,-n theorem, corresponding to l2I, C h a p . 4 ) , m a y only true in the first two categories. T h e ' c o r i e c t ' t o p o l o g i c a l L-groups could therefore be different from ours. We defin e g; (R) whenever the reference R is G-simple. This works both in the smooth änd in the pL category.

In the second part of the work we give an exact sequence similar to the C o n n e r - F l o y d neighbouring f a m i l y s e q u e n c e i n e q u i v a r i a n i bordism. This facili- tates calculations. Indeed for G of odd order we go further and show that

I I

E q u i v a r i a n t L - T h e o r y I : O S

yf (R) is a direct sum of ordinary L-groups for Q : S, ft. If R : R(.f, .f, E) ut above and G is odd then

g f ( R ) = I t 1,,,r,(u[E(x, H)], w).

Here E(.x, H) are the groups

E ( x , H ) : f t . ( E W H ( x ) X w n ( , ) ltr,t("))

where r e -füH, 1tr" (r) is the component of .At'H which contains -x and wH (x) c wH is the stabilizer of .VH(x) in n6lü1r, and n(x, H): dim ltrt(").

At many places below our arguments are based on stability results for automorphism groups of representations. We state the relevant results. refering t l r e r e a d e r to [15II, III] for details. Let v be an IRG-module with

5 ( d i m V H < d i m V K - 3

f o r i s o t r o p y g r o u p s K=11. We have automorphism groups G L o ( V ) c P L 6 ( V ) - {r,y' o V) - FcV)

where Fo(V) consists of the proper G-homotopy equivalences of Z Then A u t , t (V ) - A u t o ( Z O R )

is (dim Vc- 1) connected except for Auto:.Tol. where the map is (dim Zc - 3)-connected.

The paper is divided into sections as follows Purt I

1 . P r e l i m i n a r y Notions

2. The Equivariant Surgery Obstruction Group 3. The 7T - 7t Results

4. Functorial Properties

Appendix. comparison with other Definitions (Smooth category) Part I I

1. The Orbit Sequence

2. Decomposition of Equivariant L-Theory 3. The Rothenberg Sequence

4. The Exact Surgery Sequence

l. Preliminary Notions

Equivariant topology is burdened by an involved set of notions, designed to keep track of the combinatorial structure of the components of the various fixed point sets and their fundamental groups. In this section we collect the necessary notions, namely the equivariant fundamental groupoid, the fibre trans- port and the equivariant degree, refering to [1 2, l3f for certain details.

I

W . L ü c k a n d I . M a d s e n 506

( r . 2 )

and uct "

A groupoid is a small category in which-all morphisms are.isomorphisms' The fundamental groupoid "* it a space has objects the points of X' and morphisms lv: x0 - t r utt homotopy classes- of paths from x t to xo '

Given a finiie gräup G, let 0 (G) be the orbit category of homogeneous spaces GIH and G-maPs.

(1.1) Definition. An 0(G)-groupoid is a contravariant functor from (t(G) to groupoids. The fundamen tal O(G)-groupoid zG X of a G-space X is the functor n G x ( G lH): n Homc G lH, x).

G i v e n a n 0 ( G ) - ! r o u p o i d I a n d a s u b g r o u p H o f G , l e t E ( G 1 H ) ^ b e t h e isomorphism classes of objec ts in I (G/FI), conside red as an WH-set' For r e9 (G I H ) , l e t w H ( x ) b e t h e i s o t r o p y g r o u p o f i e E ( G : H ) ^ , a n d l e t A u t ( - x ) d e n o t e the automorPhisms of the object x'

Definition. The group E(x, H) consists of pairs (o, w) with oeAuto GIH)

!f : x --+ o* x an1o.it lrÄ in'14 (G lH). Multipliöation is the " semi-direct prod- ( o r , w t ) ' ( o r, w z ) : ( o , " o r , o f w t " w z ) '

w e e l a b o r a t e o n ( 1 . 2 ) i n t h e g e o m e t r i c s i t u a t i o n w h e r e g : n c x ' T h e n E ( G ; l H ) , \ -r c o X ; " " a ' i : X o ( x ) i s i h . . o - p o n e n t w h i c h c o n t a i n s x ' I f o : G I H

- - G l H s e n d s H t o g F I t h e n g : t ' H g : H a n d o * ( r ) : g x . H e n c e r ' r ' i s a h o m o t o p y c l a s s o f p a t h s f r o m g r t o x ; a n d o 1 ( * ) i s t h e p a t h g ' w . A p o T t i n t h e u n i v e r s a l

;;;;, f'(x) is a homotopy class u of paths ending at r' There is an action o f E ( x , H ) o n X o ( x ) b y u ' ( o , w ) :o * L t " w ' N o t e t h a t

( 1 . 3 )

and the extension

E ( x , H ) - n t (EWH (x) x w n 6vXH (-x)),

I -, n r(X' , r) -' E (x, FI) ---' WH (r) -+ 1

An 0(G)-Junctor between e(G)-groupoids is a natural transformation 'f :9o ---r9r. Let l be the category with'iht.. objects 0 and l and three morphisms' namely the identities and 0 -r 1. An 0(G)-transformation h: fo-'./t between two (,(9)-functors is an 0 (G)-functor

h : 9 6 x I - 9 r w i t h h l g n t { t } : / ; ,

w h e r e g o x l s e n d s G I H t o g o g l H ) x / . w e c a l l t w o e ( G ) - f u n c t o r s /6 a n d . l 1 homotopic if there is an 0(G)-transformation between them' This is an equiva- lence relation (reflexive since morphisms 9l^!a{GlH) are isomorphisms)' The set of homotopy classes is denoted i'go,Er](f (c). The homotopy class represented b y /'is denoted [/].

Two exampläi will illustrate the concepts' lf 9i: \ are groups ( : $roupoids w i t h o n e e l e m e n t ) , a n d G : I l ) .

I

l9o, 9 rf' (G) : Hom (ft, f)llnn (r' )'

I

E q u i v a r i a n t L - T h e o r y I 507 I f E i : n o X r , a n d f: Xo--Xr is a G-map we get.f':nG X o - r l T G X r , a n d , / o = c f , i m p l i e s th a t [/o]:lfrf i n l n c X o , r o X r l t , o ) .

A n 0 ( G ) - f u n c t o r f : 9 s - - 9 r i s a h o m o t o p y e q u i u a l e n c e i f t h e r e e x i s t s a n 0 ( G ) - functor 8: 9 r '- 9o such that the composites are homotopic to the identities.

We also need the (definitely) weaker notion:

( 1 . 4 ) Definition. A weak homotopy equiualence . f : g o - g , i s a n 0 ( G ) - f u n c t o r s u c h th a t f o r e a c h GlH, f (GlH):9o@lH) --gtGlH) is an equivalence o f c a t e g o r - ies (i.e. full and faithful, and a bijection on isomorphism classes of objects).

Given I we can compose with the "space of a category" construction [16]

t o g e t a f u n c t o r l9l:0(G)-{Spaces}. Consrruction ( C ) o f [6] gives a G-CW c o m p l e x K ( E , 1 ) together with an (. (G)-functor

such that l-t: ,rG K (9' l) 'I (i) p is a weak homotopy equivalence ( 1 . 5 )

(ii) K (8, 1), - K(E (GlH), r).

(cf. [18]).

(1.6) Proposition. ([12]) Let Y be a G-CW-complex antl p: Trcy---,g an (,(G)- .füncto^r. Then ,(Y, p) satisJies c'ondition (1.5) iJ' and only if the map Lx, yfc

- lnn x , gltr tct which sends lfl to lpt" ro .f'l is bijectiue fir itt c - CW-compleies X.

F o r G : 7 , ( 1 . 6 ) r e d u c e s

- Hom (n, X, n)llnn(n). to the isomorphism LX, K(n, I)f

( 1 . 7 ) Corollary. (i) An e@)-functor f:go-g, i s a w e a k h o m o t o p y equiualenc,e i f a n d o n l y if it induces a hijection f*:lno X,go1erct-lno X,grlo,o) .for all G - C W - c o m p l e x e s X .

(ri) For G - CW-complexes X and Y any weak homotopy equiualenc:e .from nG X to nG Y is a homotopy equiualence.

We next consider the equivariant fibre transport and its associated equivar- iant first Stiefel Whitney class. We shall work with locally linear G-lR'bundles, [ 1 ] , p . 2 1 8 ] . T h i s c o n c e p t c a n b e t a k e n in t h e smooth category ( G - v e c t o r b u n d l e s ) or in the piece-wise linear or topological categories. We assume in the two latter cases a distinguished zero section, preserved by all bundle maps. When there is no need to separate between the categories we simply speak of G - lR, bundles or G-bundles when it is unnecessary to specify the dimension.

Write ts,(X) for the groupoid of G-lR'bundles over X and isotopy classes o f G - l R ' b u n d l e isomorphisms ( o v e r idx). A G-map f: X --, yinduces a functor . f ' * : B ^ ( Y ) 't s , ( X ) . L e t t i n g X v a r y o v e r t h e h o m o g e n o u s s p a c e s w e g e t a n ( ' ( G ) - groupoid

I I

t s . n : 0 ( G ) - {groupoids}.

508 W . L ü c k a n d I . M a d s e n

G i v e n a G - l R ' b u n d l e ( I X , l e t t p a ( G l H ) : n o ( X ' ) - - t s , ( G l H ) b e t h e f u n c t o r which maps x: GIH --+ X to x* (. A morphism o: x --+ y, given by a G-homotopy o : G l H x I - - + X , i s s e n t t o t h e f i b r e t r a n s p o r t tp s ( o ) : x * ( - - ' y * ( d e f i n e d a s t h e r e s t r i c t i o n to G I H x {0} of a bundle isomorphism x*{x I--ro* ( which is the i d e n t i t y o n G I H x {1}. The collection tpa$lH) defines a n e ( G ) - f u n c t o r ( t h e .f ibre- transport of o

( 1 . 8 )

tp4: nG X --lß.n.

L e t [tpr]'. X --+ K ( t s , , 1 ) b e t h e c o r r e s p o n d i n g G - h o m o t o p y c l a s s , c f . ( 1 . 6 ) . W e have the following elementary properties of the fibre transport, where 7 is the c l a s s i f y i n g G - l R ' b u n d l e o v e r t h e c l a s s i f y i n g s p a c e B ( G , n ) :

( 1 . 9 i ) t p u is a w e a k h o m o t o p y e q u i v a l e n c e . ( 1 . 9 i i ) I f fr: X-- B(G, n) classifies ( then

[ t p r ] " l . f < f : [ t p q ] i n [ n G x , F , ] t G '

(1.9 iii) If X is a 1-dimensional G - CW complex then t s ^ ( x ) = l n o x , l B n l c t c t o t [ ( ] - [ t p < ] .

The first two properties are obvious from the definitions. The third uses t h a t i : Y - - K ( n " { 1) is 2-connected o n a l l f i x e d s e t s , a n d ( 1 . 6 ) .

In the non-equivariant situation, an lR'-bundle over a 1-dimensional space i s d e t e r m i n e d b y i t s u n d e r l y i n g s p h e r i c a l f i b r a t i o n s i n c e n r ( B O ) = n t ( B P L )

= 7 r t ( B T b p ) = n r ( B F ) . T h e corresponding e q u i v a r i a n t s t a t e m e n t i s f a l s e . I n d e e d z o ( A u t 6 VD+IV', V')o in general for representations V , c f . [20, p. 101], [15 II].

Thus we must study equivariant spherical fibrations separately.

A G-S' Hurewicz ftbration ry over a G-CW-complex X is locally linear if for each r in X there exists a G-.-invariant neighbourhood (J*c X such that qlU, is G,-fibre homotopy equivalent to a trivial G,-fibration U,. xVi for some n-dimensional G"-representation 2,.,. Here Z-'denotes the one-point-compactifi- cation. We abbreviate and call such fibrations G - S' fibrations. They have a classifying space BF (G, n).If ( is a G - IR' bundle its fibrewise one-point compac- tification {' determines a G - S' fibration.

L e t t s { ( X ) b e t h e g r o u p o i d o f G - S ' f i b r a t i o n s o v e r X w i t h h o m o t o p y classes of fibrewise G-homotopy equivalences as morphisms, and let ts,Q bc the corresponding e(G)-groupoid. Each ry:ts.F"(X) gives a fibre transport [ t p r ] e l n o x , B F n f c ( i ) .

Fibrewise one-point compactification gives a G-map J : B ( G , n ) - B F ( G , n ) ,

w e l l - d e f i n e d u p t o h o m o t o p y , a n d a c o r r e s p o n d i n g (' ( G ) - f u n c t o r j: 8 , - - t s F , s u c h t h a t [ t p r ] " l n o J f : L / ' ] , , [ t p r ] . F o r a n 0 ( G ) - f u n c t o r J ' : n G X - l B , , w r i t e . / ' ' for j " J'. In particular tp*. : tp! for a G - IR' bundle (.

( 1 . 1 0 ) Definition. T h e e q u i v a r i a n t f i r s t S t i e f e l - W h i t n e y c l a s s o f a G - S ' f i b r a t i o n 1 1 i s t h e h o m o t o p y c l a s s ** ( r y ) : [tp,] in [zG X,BF,f((G).

J

t

E q u i v a r i a n t L - T h e o r y I 5 0 9

If ( is a G -lR' bundle we abuse notation and write w(O instead of lr((').

F o r a G - m a n i f o l d M , w e u s e t h e a b b r e v i a t i o n s tp * : t p t - p r . t p ' * : t p r r . . a n d t \ ' ( M \ : w ( T M \ .

Given G-manifolds M and lü of the same dimensions and a G-map ./': M -- lü, we define an equivariant degree which collects all information about the degrees o f t h e c o m p o n e n t s o f t h e f i x e d p o i n t m a p s f", Mo --NH. This requires that we can compare the local orientations of M and l{ at corresponding points.

W e f i x a n 0 ( G ) - e q u i v a l e n c e E : . f * t p i , - - t p ' r , g i v i n g G , - h o m o t o p y e q u i v a - lences q,: Ts61-lü'-- T, M', compatible with the action of G and with fibre trans- ports along curves. The degree of .f : M -- l{ will depend on E.

Consider first the non-equivariant case G: 1 and suppose M and l{ are c o n n e c t e d . L e t _yel{ be an (interior) regular value for f. For re.l ' ( l ' ) there is a commutative diaeram

Here i ir an isomorphism (in the relevant category), i and .j are inclusions w i t h t ( 0 ) : x , / ( 0 ) : ) , & n d t h e d i f f e r e n t i a l s d i o a n d d . j u a r e t h e i d e n t i t y o f T , M and l N, respectively. Define

T , M ' ' ' n4,"

l l

t l

l ' l '

+ ^ +

M ---r---+ N

( 1 . 1 1 )

F o r a n ( t . 1 2 )

deg(J'; d: I d.g (q".,.*).

x e f I (.r')

( ( G ) - g r o u p o r d 9, write

Conj (:e):llv 1c1u)^ lwH ,

where (F1) runs over conjugacy classes of subgroups of G and where the circonflex denotes isomorphism classes of objects in the given category. Thus for g:nG l',1, '9(GlH)^

IWH:zo(Nr/)lWH. The element in Conj (9) determined by 1,e9(GlH) i s d e n o t e d (y , H ) ^ .

L e t C : N " ( y ) b e t h e c o m p o n e n t o f l { ä , w h i c h c o n t a i n s y . L e t C r , . . . , C , be the components of MH which map into C and .fi: Ci--C the restriction of .f ".Then dim C,: dim C. The given ( (G)-transformation defines

e i : J ' i * tp ä - - tp|.,, (E),: E! .

(1.13) Definition. The element Deg (./; rp)e Hom (Conj (nn lt'), Zl is given by

D " g U ' ; E ) 0 , H ) ^

on (-l', H)^ eno?llr)lwH.

dee(.f t; qi)

- s

- L

, ' - I

5 1 0 W . L ü c k a n d I . M a d s e n

We list some obvious properties of (1.13). Consider a G-map of triads F : ( P ; M , M * ) - l Q , l ü , l{ * )

with F lM:J'. Suppose Q: F* tpb- tp! agrees with (the suspension of) E when we identify T" P:T,M@lR for-xeM using the inward pointing uormal. The inclusion induces i'. no lü -- zG Q. Let/* be the Z-dual' Then

( 1 . 1 4 ) Dee(f; E):.i* Deg (F; @)

A s a s p e c i a l c a s e , c o n s i d e r a G - h o m o t o p y h : M x 1 - - + l ü b e t w e e n fo a n d . f 1 ' Fibre transport defines an ('(G)-transformation

Ü n: ./'{ tpi, -' /i* tP'".

W e g e t f r o m ( 1 . 1 4 ) t h a t

D e e ( f o , e , ' Ü n ) : D e g ( . 1 \ ; d -

Finally, suppose we have G-maps .f : L---, M, g: M'-rl{ and ( (G)-ftansforma- t i o n s

E: ./'* tP'v t tq'r., Ü: g* tPi' --+ tP'r ' If zo (gH) : no(Mlr) --+ fto(Nfi; is bijective for all H c G then

( 1 . 1 5 ) Deg (g " f; E ".f* W)): Deg (s; Ü)(g*)-' Dtg (J : q')

as functions on Conj (no lU).

Consider the partial ordering in Conj (9) given by

( 1 . 1 6 ) ( x , H ) ^ 3 ( y , K ) ^ o o * y n : ' 1 4 i n 9 ( G l H ) ^

for some o: GIH -- GlK. For any ((G)-functor f :9 '--, tsf there is a dimension function

Dim1,1 : Conj (9) -, Z

( r . t 7 )

whose value on (x, 1{)^ is equal to the dimension of the H-fixed set of the f i b r e t ( G 1 1 1 ) ( x ) " o . F o r 9 : n G l ü a n d t : t P u , D i m 1 , , ( X , H ) ^ i s t h e d i m e n s i o n of the component lüH(x) of lüH which contains x'

( 1 . 1 8 ) Definition. (i) For a fixed e(G)-functor f ' . 9 - + t s 4 , , (.y, K)^eConj (9) is called an isotropy object if

( x , H ) ^ > ( y , K ) ^ - D i m 1 , 1 ( x , H ) ^ a D i m 1 , t ( ) ' , K ) ^ . The set of all isotropy objects is denoted Iso (r).

(ii) The C(G)-functor f satisfies the weak gap conditions if for each pair of isotropy objects (x, 1l)< (y, K)

I

8 - < D i m 1 , r ( y , K ) ^ + 3 { Dim1r1(-x. H ) ^

_ l

E q u i v a r i a n t L - T h e o r y I

(iii) It satisfies the strong gap conditions if

I 0 < 2 D i m 1 , 1 ( y , K ) ^ 1 D i m 1 , , ( r , H) ^

A G-manifold l/ satisfies the gap conditions if tpiu does. Consider such G- manifolds and G-maps ,f: M --+ l{ for which no .f '.. no MII --> ftol{H is a bijection f o r a l l H c G . Under these a s s u m p t i o n s w e h a v e th e f o l l o w i n g r e s u l t s f r o m t13].

( 1 . 1 9 ) P r o p o s i t i o n . S u p p o s e . f o r e a c h p a i r o f ' c o r r e s p o n d i n g ( ' o m p o n e n t s C c M I I and D c lfH

( f ' ) * ( w , ( D ) ) : w r ( C ) a n d d e e ( f " : C - - D ) : 1 .

Then

(i) there exlsf.s a unique Deg (/'; E)= 1 in Hom (Conj (rro N),

it tf lGl ls odd, then J'or any E,

0 (G)-transJormation E: .f * tpru - tp,u witlt

z\.

D e g ( J ' ; q ) i s c o n s t a n t e i t h e r * 7 o r - 1 .

2. The Equivariant Surgery Obstruction Group

Our definition of the equivariant surgery obstruction groups is modeled upon C . T . C . W a l l ' s g e o m e t r i c approach in the case of G: l, l2l, Chap. 9]. It is a variant, and extension, of the surgery groups introduced by T. Petrie and H. Do- v e r m a n n in [3].

The basic problem in the equivariant setting is which bundle data to use.

On the one hand one needs unstable data in order to make surgeries and on the other hand one wants to be able to construct normal maps by equivariant transversality; this corresponds to G-stable bundle data. Only in special cases (e.g. lGl odd and PL category) can one destabilize G-stable bundle data. We first present our definitions and then give a discussion of their applicability.

The definition of th makes sense both in the smooth and in the locally linear PL or Vt,f categories. For .9'one needs the smooth or the locally linear P L category.

Given G-bundles (, and (r, an IR-srable bundle map from (, to (, is a G-bundle map

f : 1 , @ I R A ' - - (, 6l IR^'

where G acts trivially on lRk. It will not be necessary to keep track of the d i m e n s i o n s f t , , s o w e w i l l a b b r e v i a t e n o t a t i o n a n d w r i t e .f: (r-(2.

Given an (,(G)-groupoid and an (,(G)-functor t:g --ts4, (or IB,), its ft-fold suspensio n Zk t is the composit e I ---, ts4 -I1-+ ts4, * o. An lR-stable e (G)-transfor- m a t i o n b e t w e e n the (;(G)-functors r , a n d r , i s a n ( : ( G ) - t r a n s f o r m a t i o n q : Z k ' t , -- Zk' r, . Again we will often abbreviate and write et t y --+ t r. The (. (G)-grou- poids IBn and ts{ were defined in Sect. 1.

(2.1) Definition. A G-normal map (f,.f, d of triads consists of (i) a G-map

I

( f ; ö r . f , ö o . / ) : ( M ; e , M , A o M ) - - ( N ; ä r N , o ' o N )

512 W . L ü c k a n d I . M a d s e n

o f G - m a n i f o l d s w i t h A M : A o M v A r M , A @ t M ) : e r M a c ? s M : A Q o M ) a n d

c , f : f l a , M ,

( i i ) a G - l R ' - b u n d l e ( o v e r N a n d a n l R - s t a b l e G - b u n d l e m a p f : T M - - i over /,

(iii) an IR-stable 0(G)-transformation q: tp'tl tp;u' If A I M: @ we call (f. f, ,p) u G-normal map of pairs' A G-normal map gives the ]R-stable (t(G)-transformation

E'.f : tp'nn -' .f * tP2--./* tP';

s o w e h a v e t h e f u n c t i o n D e g ( J ; k p " . f ) - t ) : c o n j ( t t o l r ) - - + 2 . It does not matter that q..f is only given IR-stably.

I n ( 1 . 1 8 ) w e t e f i n e d t h e c o n c e p t o f i s o t r o p y o b j e c t i n !onl-(zcl{) w ' r ' t ' t p . , , ; ( r , H ) ^ e n o N H l l ' { H i s i s o t r o p l i i t t t r e c o m p o n e n t C ( x ) o f l ü H d e t e r m i n e d by r contains u.t .I.-.nt with iiotropy group H' The subset of all isotropy objects is denoted Iso (tp"). For a G-map .f : M -- lü, /*: conj ('o M)

-- b"": (,ro N) maps (x, H)" to (/(x), H)^. A G-homotopy equivalence .f :,-X --' Y b e t w e e n G - C [ 7 t o m p l e x e s d e f i n . r " u n e q u i v a r i a n t to r s i o n * - h o ( f ) e W h c ( n o Y ) i n t h e e q u i v a r i a n t W i i t e h e a d g r o u p , t 5 ; 1 0 ; 1 4 1 . T h e m a p . / i s G - s i m p l e rf

* * h G 1 . 1 1 : 9 .

(2 2) Definition. A [G-simPle]

satisfies the extra conditions ( i ) D e e ( f ; @ " f ) - ' ) = 1 ,

G-surgery problem is a G-normal map which

Iso (tpr) : fi1 Iso (tP"), G-homotopy equivalence'

(2.3) Definition. A reference R :(9, to, t t, z) of ambient dimension n consists o f a n e ( G ) - g r o u p o i d 9 , t w o e ( G ) - f u n c t o r s

t o : 9 - t l B n y 1 , . t r : 9 t I ß . n

a n d a n l R - s t a b l e ( ( G ) - e q u i v a l e n c e T , . t c o . - + s c , b e t w e e n t h e a s s o c i a t e d f u n c t o r s t , i : g - - + t s F , * u . f f " r . " u r p . n r i o n IR i s t h e ( n + l ) - d i m e n s i o n a l r e f e r e n c t I R

: ( 8 , Z t o , Z t r , Z " c ) .

( 2 . 4 ) D e f i n i t i o n . A r e f e r e n c e R : ( 9 , t s , t r , r ) i s c a l l e d G - s i m p l e if f o r e a c h p a i r ( - x , H ) w i t h . x e 9 ( G l H ) . t h e h o m o t o p y e q u i v a l e n c e

r: t'o$ lH) (x)"rt '-+ t t(G I H) (x)"tt is H-simple.

n map of [G-simple] references of the same ambient dimension ( i i ) . / . : C o n j ( n o u ) - - C o n j ( z G N ) h a s

( i i i ) e ä 0 f , A o M - - + d o l ü i s a [G-simple]

(2.s)

i s a t r i p l e p : ( i , F o , l t , ) c o n s i s t i n g o f a n ( , ( G ) - f u n c t o r ) . : 9 ' - - ' 9 ' a n d l R - s t a b l e [G-simple] e (G)-transformations Fi: ti -' Ä* ri such that

( i ) t ' , ' F l o : lt't"r

( i i ) 4 ; 1 I s o ( r " ) : I s o ( r r ) .

I

E q u i v a r i a n t L - T h e o r y I

T h e o p p o s i t e of a reference m a p p : ( Ä , Ho, lt,)is defined to be - P - ( ^ , -

H o , - F ) w h e r e - Fi:p; @ (-id): f; @ tpe

Given a G-normal map ( r r o l ü , tpq, tpr, E).

( 2 . 6 ) Definition. A G - n o r m a l map with reference R : ( g o, to, t,, z) is a G-normal m a p t o g e t h e r with a map of references p : ( z c l / , tp<, tprv , E)-(g, to, tr, r).

Given an n-dimensional reference R, we can now imitate the geometric defini- tion of L-groups given in 121, chap. 9] as certain bordism groups. Let

.f: (M, AM)--(,^ü, e.^/), ä/ [G-simple] homotopy equivalence, f: TM --+ (,

P : tp,t -- tPru.

p : ( n G l ü , t p s , tp , v , g ) - R

be a G-surgery problem of pairs with dim M:n and with reference R.

A [G-simple] null bordism of (/', i, rp, p) is a G-surgery problem of triads of one dimension higher (F, F, e, p),

F : ( P , A o P , ä , P ) - r ( Q , ö o Q . e , e), ö, I- : 1

F : T P - q , e r F : . f , ö o F a [ G - s i m p l e ] h o m o t o p y e q u i v a l e n c e r b : t p i - - r t p b , p : ( n G Q , t p * t p e , d ) - I R

s u c h t h a t t h e r e a r e i s o m o r p h i s m s u o : A r P - - M , u e : d o e-lr{ and an lR-stable b u n d l e m a p ü 6 : 4 l A o Q - ( f o r w h i c h a l l t h e o b v i o u s compatibility conditions h o l d .

More generally, two G-surgery problems of pairs over R are bordant if t h e r e is a n u l l b o r d i s m o f t h e d i s j o i n t u n i o n ( . f o , f o , e o , p o ) * ( f r , . f r , e r , _ p ) . (2.7) Definition. The bordism classes of G-surgery problems of n-dimensional p a i r s w i t h r e f e r e n c e R : ( E , t o , t r , r ) i s d e n o t e d g , ! ( : E , t o , t t , z ) . If R i s G - s i m p l e the corresponding G-simple bordism classes of G-simple surgery problems is d e n o t e d Y ; ( E , t o, t 1, r).

T h e s e t s 9 , ! ( p , t o , t r , r ) a n d y ; ( E , t o , t r , . c ) of bordism classes. The zero element

-1.f. f, E, pf :lf, i, E, - pf.

A m a p p : ( r 9 , to, t r, r) - (g', t'o, t'r, r) -+ 9^(R'), so our 9-groups are covariant

T w o m a p s p : 6, 1",o. 1 " t r ) a n d p : ( ) , there is an ( (G)-transformation ü : ) --X"

- - 2* rl@ tpn.

U,.f, d we get an associated reference

groups under disjoint union the empty 6 ---> e and

o f r e f e r e n c e s i n d u c e s a m a p p * : !)(R)

functors.

Fo, [t,) from R to R' are homotopic of such that

),* t"

J', I

I* t',

are

i s

i--.---+

" r \

5 1 4 W . L ü c k a n d I . M a d s e n

commutes for i:0, 1 where r/, is induced from r/ in the obvious way, ü , G I H ) ( r ) : t ' t Q I H) (ü G I H) (x)).

(2-8) Lemma. Homotopic maps of references induce the same map of .g-groups.

ProoJ'. There is an obvious normal bordism, by crossing a normal map with reference R with 1. n

(2.9) Remark. Let G: 1 and let g: z be a group. An c(l)-functo r t: n-- IBn is an n-dimensional vector space together with a homomorphism w: Tt--+no Aut(v)

: { t 1 } . c o n s i d e r t h e r e f e r e n c e R : ( n , t , t , i d ) . c o n j u g ä t i o n w i t h a f i x ö d g r o u p e l e m e n t g e z defines an automorphism p:(c(g),id, id) of R. This is homotopic to the auromorphism o' : (id, w (d . id, w (g) . id). Inde ed, ü: c (g) -- id is the c,(\- t r a n s f o r m a t i o n i n d u c e d b y t h e m o r p h i s m g - 1 . 8 y ( 2 . g ) , p * : o x . B y d e f i n i t i o n o f t h e g r o u p structure in 9^(R), 6*: - id if w(g): -1; cf. the discussion and slight correction of wall's definition of L-groups given in l7l.

In the next section we prove under mild restrictions on R that each element of %'t (R) can be represented by a G-surgery problem with p:(zcN, tp<, tp,,u, g) --' R such that i: nG IV -- I is a weak homotopy equivalence.

Our lR-stable bundle data used above are more restrictive than one would like them to be, so it is in order to make two points. First, they look more restrictive than the bundle data used in [3, Chap. 4], but actually they are not, see the appendix. Second, for G of odd order and in the locally linear P L c a t e g o r y the natural G-stable bundle data, f:TM@v-( ( v an urbitrury IRG-module can be desuspended to the lR-stable ones provided M (and N) s a t i s f i e s t h e s t r o n g g a p conditions, c f . [15II].

(2.10) Lemma. Let f: (M, A M) - (lü, al/) be a map coueretl by anR.-stahle bundle mup f: TM -, -(. Then there erisfs a representation W and afi.brewise G-map

A: G @ W)'-+ (Zl/ @ W), .

ProoJ.Let V be a representation in which M can be embedded. Then the compo- s i t i o n M - r l / - ( @ z i s h o m o t o p i c to a G - e m b e d d i n g , s a y .f ' o : M - r ( @ v a n d

f { ( T ( ( O z ) ) = . f ' * ( ( @ v ) @ . f * ( T l / ) = r M @ I/@/'*(rt/).

It follows that v(.6)@ TM=

may assume

V @ f * ( T l i ) @TM and (after a change of V) we v ( f d = V g 1'* (j|N).

we now collapse onto a tubular neighbourhood of fo@) in ( @ v and compose with the bundle map f 4' (m0 -- ?.lü to get a G-map of Thom spaces

ö : l t / { @ / -- M f * ( ' r N ) @ / -- 1 g 7 ' N @ /

t ^I

I

E q u i v a r i a n t L - T h e o r y I

This is induced by a fibrewise G-map

A t G @ W ) ' - - + ( T / @ W ) '

for some representation W which contains Z Indeed, embed lü in some represen- tation U and add v(l{, U) to @; this gives

lvr( o r@ u -- .lüu @v : N+ A (U @ V)'.

Since the Thom space is the quotient of the fibrewise one point compactification by the section at co, we obtain

( ( @ V @ Y ) ' - ( U @ V ) '

and we can add 7l{ again. This gives (2.8) with W:U @ V. f

The converse of (2.8) is not in general true. Given G-bundles ( and 4 over N, and a fibrewise G-map Q:('--rl'. Suppose @ can be deformed to a G-map / w h i c h i s G - t r a n s v e r s e t o l ü c 4 . S e t M : f - t ( l { ) .

T h e n f: M --lü is covered b y a b u n d l e m a p f: TM @f*(,i- T l { @ ( . T h u s w e p i c k a c o m p l e m e n t rt @ ( = N x U t o e e t

This is not a normal map in the sense of (3.1(ii)) unless [/ is the trivial representa- tion; we need to desuspend f. Second, in general @ cannot be made G-transverse, see [15I] for a discussion. If G has odd order, and N satisfies the strong gap conditions then the transversality and desuspension can always be achieved in the locally linear PL category.

T h e n e x t l e m m a is a n e a s y c o n s e q u e n c e o f ( 1 . 1 3 ) a n d ( 1 . 1 8 )

( 2 . I 1 ) L e m m a . L e t ( F , f * , f ) : ( P , M * , M ) - - , ( Q , l ü * , N ) b e a G - m a p b e t w e e n triads satisfying the weak gap conditions. Suppose the inclusions M * c. P and l{* c Q are G simply connected and that f * is a G-homotopy equiualence. Then there exisfs an 0(G)-transformation E:./* tpn, --+tpu such that Deg(f; E)=1.

Hence a G-map which satisfies 2.I (1), (ii) can be normally cobordant to a G-homotopy equivalence only if (2.1 iii)), (2.21)) and (2.2(ii)) are satisfied.

3. The n-n Results

In this section we prove (under mild restrictions) that each bordism class of G-surgery problems contains a representative whose reference map p is an equiv- alence. This is the G-quivariant version oi l2l, Theorem 9.4f: Lt,:1. Also the equivariant T-ft theorem is valid provided we work in the smooth or in the PL categories, cf. [4], or [21, Theorem 3.3].

First, we use the ordering of (1.16) to introduce a necessary restriction on t h e r e f e r e n c e R : ( 9 , t s , t y , T ) .

T M @ ( J f , T l ü @ ( ( @ 0

l l

1 t

M ---!---+ N.

5 1 6 W . L ü c k a n d L M a d s e n

( 3 . 1 ) Definition. An e(G)-functor f :9_+tsf, is called geometric if for each ("x, Il) ^ e Conj (9) there exists a maximal isotropy object (.xo, Flo) ^ e Iso (t), larger t h a n ( x . H ) ^ w i t h t h e s a m e d i m e n s i o n , i . e .

( i ) ( x , H ) ^ < ( x o , H o ) ^

( i i ) ( x , F I ) ^ < ( y , K ) ^ a n d ( y , K ) ^ e I s o ( r ) - ( r o , H o ) ^ ( ( , r ' , K ) ^ ( i i i ) D i m , , 1 ( x , H ) ' ^ : D i f f i t r l ( x o , H o ) ^

(iv) For oe Homo G lH), G lH o), o*'. Aut"ror",,r (ro) -E- Äut^ clnr (o* .xo).

The standard example (9, t):(nG l/, tp'") is geometric. Indeed, for each com- p o n e n t C o f l { H t h e r e i s a u n i q u e K : H w i t h C c l { K a n d K : G , f o r s o m e x e C .

A reference R : (9, to, t t, r) is called geometric if (9, r" ) or equivalently ('9, t'o) is geometric. Our main result is the following

(3.2) Theorem (n-n lemma). Iet R he a geometric reference rf'ambient dimension n which satisJies the weak gap conditions. Then an1, element ote g!{R) (resp. .y;(R)t c'ctntains a representatiue (.f,.f, E, p), *-ith 2: nG l',1 --r9 a tt,eak ( (G)-eEtiualenc'e.

Before we begin the proof of (3.2) we do a preliminary modification on (./,.f, E, p) to obtain the following:

( 3 . 3 ) ( i ) T h e r e i s a 2 - d i m e n s i o n a l G - C W c o m p l e x K w i t h n G K : 9 . a n d a G-map i: l,lQ) -- K from the 2-skeleton lü(2) of .^t' inducing 2 '. r'; N --'g (ll) f " : MH -- NH is 2-connected for H c G

(iii) p* : Conj (no lU)-- Conj (9) maps Iso (tp') onfo Iso (f o).

We may take K of references

t o b e t h e 2 - s k e l e t o n o f K ( ' 9 , 1 ) . B y ( 1 . 7 ) , t h e r e is a m a p i : ( n G K , i * f o , i* t t , i * t ) - + ( 9 , t s , t 1 , r ) ,

inducing an isomorphism between the corresponding L-groups, and ),'. n(; l\,1 --'.4 can be realized by 2: lvr(2) --+ K. We get (ii) by doing zero and one-dimensional surgeries.

For (iii), we add appropriate null-bordant G-surgery problems with R-refer- ence to (f, f, E, p) as follows. Let (r, H)^ e nG K be an isotropy object. There a r e t h e G - S ' * k ( r e s p . G - S ' ) b u n d l e s t' o Q l H ) ( , x ) (r e s p . fl Q l H ) ( . x ) ) o v e r G l H . We can destabilize the first one, and write

r?ä @ (R-)' - t|@ I H) (x), q \ : t \ ( G lH \ ( x ) .

The lR-stable homotopy equivalence r destabilizes to an equivariant homotopy equivalence r: ryi--tl\. The tangent bundles are TU't:p!(ry) where p' is the pro- jection for 4i. Consider the surgery problem (r. i, E, p)

T 4 | 4 P f q o

l t

4 ' o J '

4 \ ,

f

^I

E q u i v a r i a n t L - T h e o r y I

w i t h E:pT r, and map of references p:(A,id, x e 9 ( c l H ) : M a P c G l H , K ) ' W e l e a v e f o r t h e r e a d e r

5t7 i d ) , A : n G ( , x . p , ) . H e r e to check that 2 satisfies ( 2 . 5 ( i i ) ) ; ' a t t h i s p o i n t o n e u s e s t h a t R i s g e o m e t r i c .

öiu.n (.f,.f, E,p) which satisfies (3.3), we shall do zero and one-dimensional G-surgeri.r on l,: lü --t K to make ),H 2-connected, and simultaneously do surgery on J': M -- N.

One can do surgery only on isotropy components of .lü, but this suffices since we have

ß . 4 \ L e m m a . G i v e n a m a p ( i , t t o , P ) : R - - R ' o f g e o m e t r i c r e f e r e n c e s ' T h e n Ä: 9 ---,9' is a weak 0(G)-equivalence if and only if 2* : conj (9) - conj (9') maps I s o ( r o ) b i j e c t i v e l y t o I s o ( r i , ) a n d l ( G I H ) : E ( G l H ) - - 9' (G I H) induces an isomörphiim between Aut.rtnrr'(x) and Aut.u,Gtnt(i(r)) for all isotropy objects (.x, H). I

ProoJ- d' diagram

( 3 5 )

T h e o r e m - J . 2 . C ' o n s i d e r a n e l e m e n t in n 7 * r ( 2 o ) , I : 0 , 1 g i v e n b y a G - G l I I x S r ---'---.+/Y^j r , r t 2 l

G f H x D "

I t,

l k r K

We want to do surgeries on l{ and on requires that J'induces a G-isomorphism in particular that f - t (eH x St)c MH '

W e o n l y t r e a t t h e c a s e I : l ; I : 0 i s s i m i l a r b u t e a s i e r ' nience that MH and lüH are connected (and isotropic), in with connected components separately.

M simultaneously to kill (j, ft).. This from f - t (GlH x Sr) to GIH x S', and Assume for conve- general one works

[9], we may suppose that a s s o c i a t e d to t h e b u n d l e Step 1. Let V and W be the H-modules given by the fibres of ( and Tlü over t h e p o i n t k ( e H , 0 ) . W e h a v e b u n c l l e s ( o : G x r t V a n d ( t : G x n W o v e r G I H X D 2 , and, since D2 is contractible unique C (G)-equivalences k* f, -- tP(.,, s :0, 1' Using 1r., from the reference map p:()", LIo, F,) we get over GIH x Sl

Since GIH xSr is 1-dimensional we can choose (lR-stable) G-bundle isornorph- i s m s b s : j * ( - ( o @ l R k a n d b r : i * T N - ' ( t w i t h

( 3 . 6 ) t P r . : 1,.

We remark that the isotopy classes of b., are not i* ( and b6 destabilizes ft times.

d e t e r m i n e d b v ( 3 . 6 ) , a n d t h a t By the immersion classification theorems, [8],

i i s a n e m b e d d i n g o f 5 1 i n N n : { x e . A ü I G " : H I

5 1 8 W . L ü c k a n d I . M a d s e n

isomorphism bf : j* TI,{H --(f. As WH acts freely on lüo we can G-homotop / t o o b t a i n t h a t J ' H : M H - - , 1 { H is transverse to eHxSl-l{n, a n d i n d u c e s a diffeomorphism (resp. PL-homeomorphism or homeomorphism)

( f u ) - t ( t H x s t ) - - e H x S l

(cf.l2l, p. 901). After a further isotopy, we get a commutative diagram

We want to thicken i and i, and begin by destabilizing the G-bundle isomorphisms

c: i* TM -!- j* ( 3-, (o, d:b1 : j* ]"l/ --+ (,

It suffices to consider the corresponding H-bundles over S1 :eH x Sr.

Let V be an open /{-neighbourhood of St rn M which H-deforms onto S 1 . S i n c e T M I S I i s t r i v i a l s o i s T U , a n d b y G - s m o o t h i n g th e o r y [ 1 1 ] , U i s a s m o o t h G - m a n i f o l d . L e t v ( S 1 , M ) b e t h e H - n o r m a l b u n d l e o f S 1 i n U . T h e n v ( S l , W @ l R : T M l 5 1 . W r i t e V : V o @ R r ' * t a n d W : W o @ l R . S i n c e Autlr(VJ-Auts(Z) is l-connected in all three categories the maps c and d desuspend to /l-bundle isomorphisms

c : v ( S l , M ) - V o , d-: v(Sr , N) --+ Wo.

M r r ^ /

1 , 1 , _{I I}

G l H x S t i d

, G l H x S l

( 3 . 7 )

S t e p 2 . W e n e x t d e f o r m J' in a neighbourhood of MH. Let eo:Vo-+Wo be a norm preserving I/-equivariant map such that the diagram below of lR-stable maps commutes up to I/-homotopy

( v o @ n ; ' - : l e 5 ( r y . @ R ) '

,J ",1

"r)r,

ü ';f,',

(ü : E " f, cf. 2.1 (iii)). By a cofibration argument and because D e e U ; Ü- t ) ( x , H)^ : 1 we may change f up to G-homotopy so that

f : t r 1 o s o , . c : v ( S 1

, M ) - v ( S l , l r i )

I

^I

Equivariant L-Theory I

in a neighbourhood of S1. Thus there is an open G-set U in M with ( 3 . 8 ) ( 1 ) M H c ( l

( 1 1 ) J ' - I ( S l ) n U: 51 (iii) T, M' Ü , T7 6; l,{'

df ; '

> T" M, has degree 1 o n a l l f i x e d s e t s fo r x e 5 1 , ( c f . 1 . 1 3 ) .

Step j. We finish the proof under the additional assumption that (J:M in (3.8). Indeed, one obtains the desired bordism F : P -- Q by the G-push-outs:

x r r ( i d x e o )

" ( S '

I l o

I

. r ( S ' x r ( D '

I l o

I

x n ( D 2 G

G

x D V i < - G x

x s ( i d x e 6 )

X D W J + G x

x D V o ) o ' n c , M x l

I

l f x i d

I

I ' ly' x 1, x D W o ) o ' n d ,

if we identify v(,S', M) and v(St, l{) with neighbourhoods in M and .^{. The assumption U : M and (3.8 (ii)) ensure that F respects boundaries. There is an obvious extension of 2: l{ -+ K to A: Q --+ K. The bundle data extend by construc- tion and the reference map by (3.6). The map F has degree one by (1.14).

S t e p 4 . I t r e m a i n s to o b t a i n U - M . W e p r o v e i n d u c t i v e l y that U contains MK f o r m o r e a n d m o r e K , b e g i n n i n g w i t h K : H , ( 3 . 8 ) . S o s u p p o s e M ' ( c U ; w e attempt to enlarge U to Uo with (Jo= MK and without destroying (3.8 (ii)).

B y p o s s i b l y s h r i n k i n g U w e c a n f i n d a G-neighbourhood (J'of cls(U) with . / - ' ( S t ) n U ' \ L J : 6 . L e t M ' : M K \ ( U ^ M * ) a n d w r i t e . f ' : f * l M ' : M ' - ' l { K .

Note that WK acts freely on M'.

We can change

f ' x idlWK: M'lWK -- ly'K x M'lWK

relative to U' o M'f WK to make it transverse to YIWK, where Y : ( N K / l / K a H x s t ) x M ' c l ü K x M ' . T h e p r e i m a g e i s a o n e - d i m e n s i o n a l s u b - manifold of M'lWK. Two of its components can be connected by a path whose i m a g e u n d e r f'xidlWK i s h o m o t o p i c t o a p a t h i n Y I W K ( u s e ( 3 . 3 ( i i ) ) . W e h o m o t o p f'xidlWK r e l a t i v e t o M ' n U ' l W K k e e p i n g it t r a n s v e r s e t o Y I W K , and such that the preimage has one component less. In the end the preimage consists of a single circle.

We have achieved that J'': M' --+ l{K is WK-transverse to NK/NK n 11 x 51 and that (f ')- 1(^rK/^/K a H x S1; is the preimage of a circle S1 c. M'f WH under the WK-principal bundle M'-- M'lWK. Let v:Sr --51 be the map induced by f * from St c M' f WK to the circle St c ltlK IWK coming from I/K/NK n H

x S1 c l{K. Consider the 0 (Gl-ftansformation ü : @ " f )- t:

/'* tpi, --, tP'u

and let ü* b" the induced transformation on MK. Choose z e s t c l ü K o f f * . N o t i c e t h a t ( f K ) - 1 ( S 1 ) c o n s i s t s o f a p a r t a single point {y} and apartin M'. One easily checks that

a

in

regular value MH which is

( 3 . 9 ) D e g ( f * , ü \ k , K ) : d e g ( ü * 0 ) t " 7 , f *)+ lA/I(/A/ K a H l . d e g v

520 W. Lück and I. Ma<isen

!inc9 Deg(J'*, ü*)k, K) and deg (ü*o)i),.\.f *) are borh equar to one, deg v:0.

By the weak gap conditions (r.1g) and 13.3 1tiy, u *1wx --, ri^lwt<is 2-connected so that st = M *lwK is null-homotopic._H "i". tf)- 11lrrr<7lr K o H x s') is equal to wK x sr and /K restricted to e K'x sl is nuil-iomotopic.

w e c a n a s s u m e J ' * , i n a t u b u l a r n e i g h b o u r h o o d o f ' s t

: e K x s l c w K x s 1 , l o o k s li k e c x i d , : s t

. x . D ^ - I > trr) * D ^ : 1 : y r (s t , lü^). The obvious extension c x id : D2 x D^- | -- [:J x D''- ' has the propeity that its restriction to pz * g,-_ 2 does not meet Sr x l/K. Hence we can dä ,u.g.iy,

---.---J

_-__---)

(/, eJ ): (M, A M) --(t/, et/), ./': M __+ t{

This is done Il], where an involution

x: Wh(no ltr) -- Wh(rro N) s r

J I

p z

p7x

J l,^

N ( . The new normal

Tap -f * : M + + lü is equal to ./ on (J, and l * t (st) n (M*\(g o uK1)- o". we can enlarge u t o ( J o = MK such that . 7 ' - t 1 5 t ) n

U o : S r . T h i s c o m p l e t e s t h e p r o o f . f

N o t e i n particular that (3.2), applied to the normal map 6__+8, gives a n o r m a l map M --l{ with the prescribed'fundamental g r o u p o i t doto'. The next t h e o r e m w a s p r o v e d in [4] when nr(Mr):0 and in l2l,Chäp. 3l when G:1.

( 3 . 1 0 ) T h e o r e m ( n - n t h e o r e m ) . I e t . f ' : ( M , C o M , A , M)-(N, dolf, d, ly)be alG_

simple) G-surgery problem in the imooth ir pL.ui.gory. duppur. 'n,16r-lv) -- zo(lt') is a weak homotopy equiualence ancl that Il satisfies the strong gap conditions (1'1s). Then F is G-normally cohordant (rel 010l/) ro a surgery pr1blem ( / * , Co J ' * , i ' 1 J ' a ) ** h i c h i s a l s i m p t e f G-hctmotopy e q u i u a l e n c e o f ' t r i a d s . I

W e s h a l l not elaborate on the proof of (3.10). It is similar to the proof presented in [4], but two comments are in order. Firstly, there are certain minor errors in [4], which the reacler can overcome by using some of the material presented in the proof of (3.2) above. Secondly, in the extension of the 7T - 7r theorem to the case of non-simply connected fixed sets one uses the group e x t e n s i o n .

| '-+ ftt(M', -x)- E(r, H) - WH (x) --+ 1

and the E(x, -H)-action on the universal covering tw'(t)- over the component M'(x) of MII which contains x. The surgery I don. in the regular part of M H ( 9 - ' In the G-simple case one needs to compare the equivariant Whitehead t o r s i o n s wh(J) and w h(.f, A.f) for G-homotopy equivalences

I

^I

E q u i v a r i a n t L - T h e o r y I

is constructed (by reversing /r-cobordisms) such that ( 3 . 1 1 ) w h ( . f ) : - * w h ( f . ö f ) . (Here we use the assumption that R be G-simple.)

I n p a r t i c u l a r , w h ( f ) : 0 + w h ( f , A . f i : 0 . S e e [ 4 , C h a p . 5 ] f o r a d i s c u s s i o n w h e n n r ( l ' { " , x ) : 0 f o r all H c G, xe lür1.

Finally, one defines the equivariant surgery obstruction of an n-dimensional norm_al map (f, f. 0 to be the class )o ( f, f, ,p) it represents in 9,(ro r^V, tp(, tpru, E). From (3.2) and (3.10) one gets

(3.12) Theorem I.f N xI satisJies the strong gap c'onditions in (1.18), then a (G- simple) normal map (J',f, q) is normally c:ohordant to a (G-simple) G-homotopy equiualence i/' and only if' AG ( f', f, E):0.

4. Functorial Properties

This section compares the G-equivariant L-groups for varying G. Let i: f --+Q b e a h o m o m o r p h i s m of groups. I t i n d u c e s a f u n c t o r i: ( ( f ) - ( , ( G ) , i( X ) : G x , X.

For any category (€ consider the functor categories [(K(f),61, of contravariant functors. Then i induces

i * : l ( ' ( G ) , ' ( ; f ' - - , l ( ' (f ) , 6 l ' . l J n d e r m i l d r e s t r i c t i o n s o n ( ; , i * h a s a l e f t a d j o i n t

i*: l(t (f). r6'1'v --+ l(t(G), ,€l',

( s e e e . g . [ 1 7 I I , $ I 7 ] ) . I n p a r t i c u l a r , f o r e a c h I e l('(G), %lp there is the adjunction ü V ) : i * i * I - - + 9 .

We spell out the definition of i*(9) when '6 is the category of groupoids. Given a n ( ( t ) - g r o u p o i d 9, and GlLe ((G)let

( 4 . 1 )

(4.2) ,9(GlL)' : LI .9[lK)x Hom,, (GlL. iIlK)).

I l K e ( . t ( l\

There is an equivalence relation on 9(GlL)'. defined as follows. For a f-map /: I- lK --+ f lH and object

(u, u)eE (f lH) x Hom6, (G lL, i I lK))

we set

Similarly for a

('9 (f ) u, u) - (u. i(.f ) u).

morphism e'. uo + u t in '9 V lH),

(E ( 1 ) kil, u) - (8. i( f) (u))

522 W . L ü c k a n d L M a d s e n

This generates an equivalence relation and (i* E)(GlL) is the groupoid of equiva- lence classes. Letting L vary over the subgroups of G we obtain the 0 (G)- g r o u p o i d i * ( 9 ) .

If X is a f-space and Y a G-space then

i * n r ( X ) : l r c ( G x r X ) , i * n o ( Y ) : n ' ( Y ) .

Let IB,G be the groupoid of G-lR" bundles over the orbits GIH considered i n $ 1 . I t i s o b v i o u s t h a t i * t s f : B f , a n d a n e ( G ) - f u n c t o r t: 9 - - t s f i n d u c e s an 0 (f)-functor i* t'. i* I --+ tsf. An e(f)-functor /: I --,1ßi1, induces an 0 (G) func- tor by setting

i* (r): i*(:9) -- i * tsf : i* i* tsf --q. ts"G

with r/ from (4.1). If t: tp* is the transport of a f-bundle over X then i * (r) i s t h e t r a n s p o r t o f t h e G - b u n d l e G x, ( over G xrX.

Given a f-reference R: (9 , t o, t r, r) one gets a G-reference l* R : ( i * 9 , i * t o , i * t r , l * z ) w h i c h i s g e o m e t r i c i f R i s . S i m i l a r l y , a G - r e f e r e n c e S gives a f-reference i* S, geometric if S is.

If f : M --+ ly' etc. is a f-surgery problem with reference R then we can apply the functor G x.(-)to obtain a G-surgery problem with reference i* R. Similarly, a G-surgery problem with reference S can be composed with i: f -- G to define a f-surgery problem with reference i* S. This defines homorphisms

(4.3)

i*: 9,(f, R) --+ 9,(G, i* R), i * : 9 " ( G ; S ) - g " [ , i * S )

over G for a given geometric reference We next define a Mackey-functor

R of ambient dimension n. On obiects.

(4,4) GIH -- g"(H, t(H)* R)

w h e r e i ( H ) : H - . G i s t h e i n c l u s i o n . C o n s i d e r a G-map o: GIH --GlK. Choose g e G w i t h o ( e H ) : g - t K . L e t c ( g ) b e c o n j u g a t i o n w i t h g , i . e . c ( g ) ( g ) : g g g - t . We obtain a commutative diaeram

Consider the functors id and c(g): C(G)--,0(G) induced by induction with the corresponding group homomorphism. For any object GIH in ()(G) we have a m o r p h i s m r ( g ) : G l g H g - r > G I H w h i c h s e n d s S@HS-t) to ggH, and get a natural transformation r(g): c(g)--id. It induces an isomorphism of references p@): R --+ c(g)* R. The adjunction between l* and i* carries over to references to give a bijection

a d : H o m ( c ( g ) * t( H ) * R , i ( K ) * R ) - H o m ( i ( 1 1 ) * R , c ( g ) * t( K ) * R ) . 6 c ( ü

, 6

1 1

i r t t r I I i r r r

t l

' c t s ) ' K

I

Equivariant L-Theory I SZI

S i n c e c ( g ) * , ( r K ) * R : l ( I 1 ) * c ( g ) * R w e g e t an elemenr i(H)* (p(g)) in H o m ( , ( 1 1 ) * R , c ( g ) * i ( K ) * R ) . L e t p ( d : c ( g ) * i( H ) * R - - i ( K ) * R b e i t s p r e i m a g e under ad. Now define

( 4 . 5 ) o * : 9 n ( H , i ( H ) * R ) - g " ( K , t(K)* R) as the composition

g^(H,i(H)* R)--jg\ .y,(K, c(g)* i(H)* R) v"(K'r'ts)),

Y,(K,i (K)* R).

Similarly, let (4.6)

be the composition

o* : 9"(K, i (K)* R) - g"(H, t (H)* R)

y,(K,, (K)* R) ---14:-_- .9,(H , c (g)* t (rK)* R) -i- .g.(H , i(H)* c (g)* R) t ( r r ) * p ( s ) - ' ,

. g , ( H , r ( H ) * R ) .

We have to show that these definitions are independent of the choice of which g we pick with o(eHl: g- t K. This follows from

(4.7) Lemma. The homomorphisms c(g)* and g,(G, p(d) from g,(G, R) to Y,(G, c(g)* R) are equal.

Proof. Suppose thatg:TEG K. An elementin 9,(G, R) is represented by a surgery problem f : M -' lü and a reference map 7: Ir./ -- K, suppressing the bundle data.

Then c(g)* applied to this element is represenred by c(d*f: c(g)* M--+c(g)*lü and c (d* 1: c(g)* 1g -' c(B)* K. Multiplication I (d: K --+ c(g)* K is a G-map, and y , ( G , p ( i l sends the given surgery problem to f:M--+1,{,1(d"2:.1ü*c(g)*K.

These two elements agree in 9"(G,c(g)* R) since the following diagram com- m u t e s :

M f r l v

I t \ , r

r e : l

1 , , r , - ) , . ( g ) * K

r l , "

J -'* . J "'1ßY 2 c ( g ) * * c ( g t * J

, . ( g j f N

The reader can easily verify that the maps in (a.5) and (4.6) give the functor in (4.4) the structure of a Mackey functor. In particular we have

(4.8) Corollary. The equiuariant L-group g,(G, R) ls a module ouer the Burnside r i n g A ( G ) .

I

< ' l A J L +

Appendix

Comparison With Other Definitions ( Smooth Category )

W . L ü c k and I. Madsen

the bundle data required for (2.1 (ii)). We examine rhe differ- In the original source for G_surgery groups [3]

a G-normal map is somewhat difierent from ourJ ence.

F o r any G-bundle ry over a G-space X,let ,r,, be the fixed point b u n d l e over XH and write ry u for jhe quotieni bundle, trt I i\lri,, ,ufr'u ,lmilar notation for maps' Note that 4 | XH = rtfu @ q o, but with no preferred isomorphism (unless we have an invariant inner metric o-n d.

Dovermann and petrie's bundre data for a normal map over ./: M-+ ry' con_

s i s t s o f b u n d l e isomorphisms

( A 1) a : T M @ V -+ J'x (O O V ( G - m a p ) bnt TM n -.f * Gd (t/H_map)

s u c h th a t Qs:bn@ Vo and such that the ba satisfy obvious compatibility c o n d i - tions. Here V is any (large) IRG-module wiiir Vo +0.

(42)

Proposition' suppose f: M -+-l/ is a map between G-manifolds couered by a bundle map a. IJ'there exist bundre

!op, hu satisfying (Ar.)iiy there e-xisfs a hundle map f: TM @ VG --,.1*(0 @ Vo sä tiä o: I @ V..

^ Before we give the easy proof of (A 2) ret us remark that a bundre map .f : ry Iso (r7 , - 11, between two bundlei over M is equivalent to a section of the G-bundle t, 4z) ovet M whose fibre over xe M is ihe set of isomorphisms Iso (rl t,, rlz") w i t h the induced action of G,.

Proof of ( A2 )' The proof will be by induction over the orbit types. This proce- dure is described for example in [2, $s.1]. It reduces the proof to the following special case. Consider

a l M H : T M I Mu @ V -.f* GI Mr) g V

as an l/H-isomorphism. Let (MH)'be the singular set in M' consisting of points with larger isotropy groups. Suppose

a l ( M o ) ' : a r @ Vr;

for some l/I1-bundle map

a , : T M l(Mu)'@ VG --,.f'* 1 ( M r , ) @ V c We want to show that there exists an Nf/_bundle map

a r : T M I M ' @ V . - . f * 4 M " @ V G s o t h a t a l M H : a z @ V c .

I ^I

E,quivariant L-Theory I 525 The assumptions (A 1) implies that there are decompositions T M I M H = T M H @ T M u , . f * G ) = f * ( ( u ) @ f * ( i o ) s o t h a t

a l M I t : e l t @ b s @ V , V a @ V { : V d -

Hence it suffices to show that aH:4o @ Vf for some NH-bundle map ao: TMH @ vt -- .f * ( -(H) @ vG

W e d o h a v e a o d e f i n e d o v e r ( M " ) ' , n a m e l y b y s e t t i n g a o l ( M " ) " : o ' 1 . T h e l ü H - space MH is build up from (M') by attaching free l{HlH-cells, and we can attempt to extend ofl to ao, cell by cell. Suppose ao already defined over X a n d l e t y : X v l , { H l H x D i * 1 . The obstruction to extend rls over Y such that

a o @ V ä ' : d H li e i n

I ( G L H ( T " M ' @ V G ) - ! - - + G L u ( T ' M n @ V n ) )

where E adds the identitY along Vä

S O

There is no H-action on T" Mn @ V"

G L r r ( T , M H @ Hence rp is (dim MII + dim VG - homotopy group vanishes. This inductive step. I

v I I \ = o ( d i m M H + d r m

V \ .

We have used the definition (3.5') in [a] as (A 1). There is a weaker definition ( 3 . 5 ) i n [ 4 ] ( s e e a l s o [ 3 , S e c t . 3 ] ) , w h e r e a : T M @ V - . 1 ' * ( f o r s o m e G - b u n d l e ( and h r: TM s - 4 n fär some bundle functors 4s such that certain compatibility conditiols hold. However, the argument above shows that one can destabtlize ( to (' @ Vr;for appropriate {' anJthat one can destabilize a to .f : TM @ VG -' -(' a l s o in t h i s c a s e .

References

I )-connected, atrd since dim VG * 0 the stated proves the existence of a6 and completes the

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D o v e r m a n n . K . - H . , R o t h e n b e r g , M . : A n a l g c b r a i c a p p r o a c h to t h e g e n e r a l i z e d W h i t e h e a d g r o u p ' I n : J a c k o w s k i , S . , P a w a s o w s k i , R . ( e d s . ) T r a n s f o r m a t i o n g r o u p s ' P r o c e c d i n g s ' P r o z n a i n l 9 t t 5 ' ( L e c t . N o t e s M a t h . , v o l . 1 2 l ' 7 , p p . 2 4 4 - 2 1 1 ) . B e r l i n H e i d e l b e r g N e w Y o r k : s p r i n g e r 1 9 8 6 E , l m e n d o r f , A . D . : s y s t c m s o f f i x e < l p o i n t s e t s . T r a n s . A m . M a t h . S o c . 2 7 7 , 2 7 5 2 8 4 (1 9 8 3 ) F a r r e l l , F . T . . H s r a n g . w . c . : R a t i o n a l L - g r o u p s o f B i e b e r b a c h g r o u p s . c o m m u n . H c l v ' 5 2 . 8 9 1 0 9 f i 9 7 7 \

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H i r s c h . M . W . : l m m e r s i o n s o f m a n i l o l d s . T r a n s . A m . M a t h . S o c . 9 3 , 2 4 2 2 ' 7 6 ( 1 9 5 9 ) 6.

7 . 8 . 9 .

526

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I ^I