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The Involution on the
Equivariant Whitehead GrouP
123
F R A N K C O N N O L L Y
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( R e c e i v e d : 2 7 S e P t e m b e r 1 9 8 8 )
A b s t r a c t . F o r a f i n i t e g r o u p G w e d e f i n e a n i n v o l u t i o n o n t h e e q u i v a r i a n t w h i t e h e a d g r o u p g i v e n b y r e v e r s i n g t h e d i r e c t r o n o f a n e q u i v a r i a n t h - c o b o r d i s m . I t t u r n s o u t t h a t t h e i n v o l u t i o n is n o t c o m p a t i b l e w i t h t h e s p l i t t i n g o f t h e e q u i v a r i a n t W h i t e h e a d g r o u p i n t o a d i r e c t s u m o f a l g e b r a i c W h i t e h e a d g r o u p s ' c e r t a i n c . r r e c t i o n t e r m s i n v o l v i n g t h e t r a n s f e r -u " p , o i t h e n o r m a l s p h e r e b u n d l e s o f t h e v a r i o u s f i x e d p o i n t s e t s c o m e in . H o w e v e r , i t t h e g r o u p h a s o d d o r d e r . th e s e t r a n s f e r m a p s a l l v a n i s h ' w e p r o v e a d u a l i t y [ o r m u l a f o r a G - h o m o t o p y e q u r v a l e n c e ( / , a : / ) : ( M ; i M ) - ( N , l l v ) r e l a t i n g t h e e q u i v a r i a n t w h i t e h e a d t o r s i o n o f I a n d
U . a l )
K e y w o r d s . E q u i v a r i a n t W h i t e h e a d g r o u p , e q u i v a r i a n t W h i t e h e a d t o r s i o n ' i n v o l u t i o n ' e q u i v a r i a n t h - c o h o r d i s m . d u a l i t Y fo r m u l a '
0. Introduction
Let G be a finite group. we study the involution on the equivariant whitehead group of a smooth G-manifold given by reversing the direction of an equivariant h-cobordism' This involution does not typically pi.r.ru. the splitting of Whf(M) into n?n- equivariant groups. But we show it does preserve the splitting when G has odd order' we also give a general formula for it, and use this involution to compute the whitehead t o r s i o n o f a G - h o m o t o p y e q u i v a l e n c e o f p a i r s ( . f , l l f \ : ( M , i . M ) - ' ( / v , fl{) from that of
f : M -- l{, if M and l{ are G manifolds'
Here are a few more details. The equivariant whitehead group whG(N) of a G-manifold N splits into algebraic Whitehead groups
W h c ( N ) : O O W h ( z 1 ( E W H ( C ) x * n r c r C ) )
( H ) ( ' € t t o { N I r ) w H
w h e r e W H ( C ) i s t h e i s o t r o p y g r o u p o f C e z o ( N H ) u n d e r th e W H - a c t i o n ' L e t W h ! ( i V ) be the direct summand in WhG(,V) corresponding to those components Ce zo(NH) which contain an element x e C with isotropy group G" : H. Then any element in Whi(,V) can be realized as the Whitehead torsion of an equivariant h-cobordism over N, provided that certain codimension 3 conditions are satisfied' Hence' we can define a n i n v o l u t i o n * : W h f ( N ) - - W h t ( . N ) by reversing the direction of h-cobordisms (see Section 2).
I
124
,{
F R A N K C O N N O L L Y A N D W O L F G A N G L Ü C K T h e r e is a n a l g e b r a i c i n v o l u t i o n o n e a c h o f t h e s u m m a n d s i n t h e s p l i t t i n g o f W h t ( N ) coming from involutions on the group rings. There are some places in the literature where it is claimed that * corresponds under the splitting to the direct sum of these involutions. But this is false. We do show that this is true if G has odd order (subject to a mild condition). See 4.2.Ingeneral, the involution on the split Whitehead group looks like a triangular matrix. Its entries on the diagonal are the algebraic involutions described above. The other entries are given by transfer homomorphisms associated with the spherical normal bundles of the various fixed point sets. We show that these transfer maps are nontrivial even for G : Zl2Z.
Consider a G-homotopy equivalence of G-manifold pairs U, Af ): (M, A M) -' (lü, f N).
We will prove a formula:
ro (f) - - * ?G( f, afl - or(rc(N, atf )),
where Or(/o(lü, dN)) is a correction term depending only on the equivariant Euler characteristic XG(N,Alü) and certain G"-homotopy equivalences Ex: STM* -- S?'Nr, associated with/ for any x e M. We show that <D, is zero if G is a product of a group of odd order and a 2-group and T M ,and f l{1, are linearly G,-isomorphic for any x e M.
This formula is an important tool in the proof of the equivariant z-z-theorem in the simple category.
We have chosen to work in a smooth context. A simple group, Wh["n'r114;, parametrizing topological G-h-cobordisms is defined by West and by Steinberger in I I 7 ] ; t h i s g r o u p h a s a n a n a l o g o u s i n v o l u t i o n . I n [17], the group we are using is denoted WhPoL'?1tvt). We should also mention that results analogous to those here hold when G is a compact Lie group.
l. The Transfer Homomorphism
Let WhG(X) be the equivariant Whitehead group associated with the finite G-CW- complex X (see Illman [6]). Consider a G-O(n)-vector bundle p(O: ( J X and sphere bundle p(SO: S( 1 X. Then D( and S( carry the structure of finite G-CW-complexes, unique up to simple G-homotopy equivalence, by the equivariant triangulation theorem (see Illman [7]) and we can define transfer homomorphisms,
l . l p ( S i ) * : W h c ( x y - - W h c ( S o p ( D O * : WhGlx; -' WhG(Do
as follows. They send an element in WhG(X) represented by the torsion rG1.1; of a G - h o m o t o p y e q u i v a l e n c e f : Y - - . X t o r G ( f o ) , w h e r e . f o it t h e b u n d l e m a p g i v e n b y the pull-back construction. If p(SO* and p(D0* are induced by the projections we want t o s t u d y t h e c o m p o s i t i o n s p ( D O * p ( D Q * a n d p ( S f l * p ( S 0 * : Whc(X) - WhG(X). We start by collecting some properties of the Whitehead torsion. Proofs can be found in D o v e r m a n n a n d R o t h e n b e r g [ 4 ] , H a u s c h i l d [5 ] , I l l m a n [ 6 ] , a n d L ü c k ! 3 1 .
,***","d
THE INVOLUTION ON THE EQUIVARIANT WHITEHEAD GROUP
I . 2 . A D D I T I V I T Y
L e t ( X , , X o ) b e a p a i r o f f i n i t e G - G W - c o m p l e x e s a n d i : X o' X rbea cellular G-map' Denote by X the finite G-CW-complex given by the G-pushout' Define (Yt, Yo)' j: Yo -- Yzand Y similarly. Let kr: Y, + Y be the obvious map for i : 0' 1' 2' Consider
a pair of G-homotopy equivalences (/r ,fJ:(X,,Xo) - (Yr' Yo) and a G-homotopy equivalen ceJ'r: Xz - Y, such thatfri: ifo.Letf : X -. Y be the G-map given by the G-push-out property. Then f is a G homotopy equivalence. (see, e'g', [13]' Lemma 2.13). We have:
r G U ) : k r . r o ( f r ) + k * f ( f r ) - ko.to(f)' 1 . 3 . C O M P O S I T I O N F O R M U L A
r G @ f i : f ( s ) + g * t o ( f ) ' I . 4 . P R O D U C T F O R M U L A
If X is a G-space let {Gl? -- x} be the set of G-maps x: GIH -+ X for G = H' we call x: GIH -- X and y: GIK -- X equivalent if there is a G-isomorphism o: GIH -' GIK satisfying yo =o x. Let {c 1t -- x} I - be the set of equivalence classes and (lG (x) be the free abelian group generated by'{Gl? -- x\l-.If xo(*)is the path component of XH conrainin g x(eH)we obtain a bijection {Gl? -- X\l - -- lL wtro-(X.o.)/WH sending the c l a s s o f - x to t h e . l u * o f X H ( x ) . In p a r t i c u l a r , ( l G ( X ) i s O H o ( X u ) ' H ' L e t W H ( x ) ( r e s p ' NH(r)) be the isotropy group of Xo(*)e no(XH) under the WH-action (resp' NH-action). If X is a finite G-CW-complex, define the equivariant Euler characteristic' xG6\e LlG1x1,by assigning to fx: GIH -- xl the ordinary Euler characteristic'
7(xII (xllwH(x), x"(t) a X' fl/wH1x1;.
W e g e r a n a t u r a l p a i r i n g U o ( X ) 8 W h o ( Y ) - Whc(X x y) by sending lr: GIH -- XfE ro(g)to (r x id)*rG(id x 9)for a G-homotopy equivalence g1: Y' -+ Y' T h e n w e h a v e fo r t w o G - h o m o t o p y e q u i v a l e n c e s f : X ' - - X a n d g " Y ' - ' Y :
1 . 5 . r G ( . f x s \ : x G ( X ) I t c ( g ) + t o ( f ) I x G ( y ) '
1 ' 6 S P L I T T I N G I N T o A L G E B R A I C W H I T E H E A D G R O U P S
The equivariant Whitehead group splits into algebraic Whitehead groups as follows' For G = H define (f0: ünl ieWff x *n Xt) -' WhG(X) as the composition' w h 1 ( E w H x * n X H )
( 1 )
, y g 6 w H ( E w u x X H ) ( 2 )
, w h * n ( x o ) ( 3 )
' 1 4 7 6 N n ( x " ) 9 ' w h c ( G x * r X o ) ( s )
, W h G ( X ) ,
where (l ) is given by the pull back construction' (2) by the projection' (3) by restriction'
125
br."-***,.**-
D
for
126 FRANK CoNNoLLY AND woLFGANG LÜCK
( 4 ) by induction and (5) by the map G X *"Xt -- X sending(g,x)to g.r. We obtain an i s o m o r p h i s m .
O ,tat: @ wtrllewu x *rXo) -- whc(X).
( H ) ( H l
lf Z is a space whte) is isomorphic to Own(zftr(c)), where c runs over no(Z).
Hence, we get an isomorphism
O Wh(Znr(EWH(x) X *n,,,Xo(x)))-- Whc(X).
i G l ' t - X ) l -
PROPOSITION 1.7. For any G uector bundle ( on a fnite G-CW-complex X,p(DO*p(D}* is the identity on WhG(X).
Proof. We may as well assume X is a finite G simplicial complex. Let f . X' - X be a G - h o m o t o p y equivalence. B y 1 . 3 , w e h a v e
p ( D O * p ( D { ) * ( r G ( f ) ) : p ( D { J * ( f ( f * ) ) : f * r c ( p @ f * o + r c ( f ) _ r c ( p @ 4 ) ) . Hence, it suffices to show f (p(DO): 0 for any bundle (. Because of the local triviality of {, 1 .2 and 1.4, this reduces first to the case when X is G-contractible, and then to the
obvious case p: DV -- {*} for a G-representation Z. I
T h e p a i r i n g in 1.4 induces a pairing A(G)g WhG(X)_- Whc( X) rt we idenrify the B u r n s i d e ring ,4(G) with uc({*}) (rhe map sends lclHl to x:GlH -{*}). Let
eo(x)e A(G) be the image of f 1x)e uc(xl under pr*: u\x; --+ uc({*1; : zic;. rr
0@,n) is the number of cells of type GIH x D" in X we have
e c 1 x 1 : 7 I t- t).p(H,n).lclHl
I n > - o
i n A ( G ) . F o r m u l a 1.5 above now reduces t o ( n ) * f ( f x l r ) : r " ( f ) . e G ( y ) . We derive from 1.4:
PROPOSITION | 8. If ( I x is the triuial G-o(n)-uector bundle X x p ( s O * p ( s 0 * : whc( X) - whc(x) is multiplication b y e c 1 s v 1 .
P R o P O S I T I O N 1.9. Let ( I x be a G-o(n)-uector b u n t l l e a n d f o : s f * ( - , s(
hy the pull-back construction applied to a G-homotopy equiualence f : y --+ x finite G-CW-complexes, then we haue (fu)*p(s/*O* : p(sO*f*. similarly.for hundle, we haue: (fu)*p(Df*O* : p(DC)*f*.
Proof'. Follows directly from the definitions.
In Lück ll2f, the equivariant (unstable) first Stiefel Whitney class w( is defined any locally linear G-S'-fibration S(.
P R O P O S I T I O N 1.10. Let ( and 4 be G-O(n)-uec:tor b u n r l l e s o u e r X w i t h w ( : w r y . T h e n p(SO*p(S O* and p(Sry)*p(S r)* agree.
ProoJ'. This follows from the algebraic description of p(Sfl* given in Lück[13]. An V then
be giuen between the disc
b3-**-**.-
T H E IN V O L U T I o N o N T H E E Q U I V A R I A N T w H I T E H E A D G R o U P 1 2 7 alternative proof uses the notion of an equivariant Eilenberg-Maclane space i n t r o d u c e d i n L ü c k I l]. Let ) I BF(G,n)be the classifying G - f i b r a t i o n a n d B F ( G , n ) t h e classifying space for locally linear G-S'-fibrations. Let b(0 and b(fi'. X --+ BF(G, n) be t h e c l a s s i f y i n g m a p s f o r ( a n d q . L e t i: X --+ K ( n o X , p , 7 ) a n d 7: BF(G, rr) --' K ( r o B F ( G , n ) , p , 1 ) b e t h e c a n o n i c a l G-maps. We can interpret w( and w'4 as the G-homotopy classes of the G-maps
K ( n o X , p , l ) - - + K ( z G B F ( G , n ) , 1 t , l )
i n d u c e d b V jb ( O a n d jb ( r ) . B y assumption we can find a G-map k:K(nGX,p,1)- K ( n o B F ( G , n ) , l t , l ) r e p r e s e n t i n g b o t h w ( a n d w 4 . C o n s i d e r t h e G - h o m o - topy pull-back,
K ( n o
i. 1
X , p , l )BF(G, n)
i ,
K ( r G B F ( G , n ) , l t , 1 ) .
S i n c e j b ( O - Gki, there is a G-map a(O: X-- Z satisfying koa(O =.ob(_Q and j o a ( O - oi.Let ( : (ku)*2. By 1.6, i* and j* are isomorphisms o f W h i t e h e a d g r o u p s .
From Proposition 1.9 we get:
i * p ( s 0 * t ( s 0 * i * ' : U )*a(O*p(SO*p(SO*,;' .
: (i o)*p(s(0)*p(s(0)* a((t*(i) - 1 - (j ol*p(s(O)*p(s(0)-j;i.
This is also true for 4 so that i*p(SO*p(SO*i* t : t*p(Sd*l(Sry)*t* t holds. But i* is an
i s o m o r p h i s m b y 1 . 6 . f
Remark. In the above argument (only) we make use of WhG(X) for an infinite G-complex X. This is defined exactly as in [6] bV means of strong deformation retractions Y -- X, with the modest adjustment that we require only that Y-X have finitely many cells. (See Lück [l 3] for a full treatment.)
We also make use of the transfer p*'. WhG(X) -- WhG(S(O) for a locally linear G-S' fibration (; the definition is analogous to that in 1.1, but the details are given in [l3].
P R O P O S I T I O N l . l l . Let ( and 4 be G-O(n)-uector b u n d l e s o u e r X . S u p p o s e t h a t G has odd order, and the nonequiuariant first Stiefel-Whitney c/asses w,(O and w J r y ) e H [ ( X ; Z | 2 Z ) agree. Suppose also that for any ,x€ X,,S(, =u. ST". Then w C : w q .
Proof. See Lück ll2). I
T H E O R E M 1 . 1 2 . L e t ( I X be a G-O(n)-uector b u n d l e w i t h t r i u i a l * ,(O e Ht (X ,Zl2Z1 Assume G has odd order and that there is some G representation V such that S(, = c*SZ, . f o r a n y x e X . T h e n p ( S O * p ( S f l * : W h G ( X ) 'W h G 6 ) i s ( 1 - ( - l ) ) - i d ,
Proof. Because of Propositions 1.10 and 1.11, we can assume that ( is the trivial 'G-O(n)-vector bundle X x Z. Since G is odd, VlVo is of complex type so that
Lr."*.-.
128
e G l s v l e , a ( G ) i s ( l Next we consider equivalence,
S( -l---Sr
l - r '
l l
v f +
X -!--,Y
FRANK CoNNoLLY AND woLFGANc I-ÜcT
+ (- l)armtsr'), .lclcl.Now apply proposition l.g. I G-o(n)'vector bundles ( and q over X and a G-fibre homotopy
Define a homomorphism <Do: uG( Y) - whc(sa) as follows. Any base element of uo (y) can be represented by y:fx for some x:GlH __+X. Let <Dr([y]) be the image of rG@ lx*S(: ,x*S( .- y* Si4) under (y )*: Whc(y*Sry) _- Whc(Sa).
THEOREM 1.t3. f @): p(Sry)*(IGUD + aFec(y)).
Proof. Write F as the composite S(
fo.tolFr) from 1.3. By definition, ,o(-foi O.(Xo(I,)) from 1.2.
F {
:-) f*s4 14
sry. Now rG@): rc(f#) + : p(Sq)*rc1f7 and we derive fo.rolF:'r1 : coRoLLARy r.r4. suppose that G has odd order and wr(q)e Hr(y;zr2) uanishes. I Assume there is a G-modure v with s(ryy) -o, s( v)for ail y e y.rf xo(y)e (JG(y)is zero or if (' and ,vv ar€ linearly G,-isomorpir, yo, any xe X, then p(s(ry))* (r + ( -t),)ro(f). (ro(f)):
Proof' Apply Theorems l.r2 and r.13 and the fact that any G,-map s(, -sryr, is G"-homotopic to one induced by a linear isomorphism, as A(G,)*: ttl) holds; in
e i t h e r c a s e . { D p ( x o { r ; ; : 6 . r r ' u ' r r ' a o n r \ - f v ' r
, - . . .
2. The Involution on the Equivariant whitehead Group
Let M be a G-manifold, i'e'' a smooth compact manifold possibly with boundary on which G acts smoothly. Denote
M a : { x e M l G , : H } , M ( u ) : { x e M l ( G , ) : ( l l ) } a n d M @ t - {xe M l(G") > (H)}.
The isovariant Whitehead group is defined by
2.1. whr:.(M): g wht(MHlwH).
Here wht (M HlwH) means the whitehead group of the compact manifold obtained by removing an open regular neighborhood of M,HlwHfrom MH lwH.
An (isovariant) /r-cobordism (w,M,N) is a G-maniford w with boundary 0w : M v N,such that 0M : M o N: dN and the incrusions M __+ w and N __, w are (isovariant) G-homotopy equivalences. we define the isovariant whitehead torsion rß'(w'M'lv) of an isovariant /r-cobordism (w,M,r/) inductively over the number of o r b i t types (11) with /ie Iso M - { H | M u * ol'Let (H)be maximar among these.
-rF"**,
T H E I N V O L U T I O N O N T H E E Q U I V A R I A N T W H I T E H E A D G R O U P ' f
h e n M , , , , : , y { r r 1 , a c o m p a c t G - s u b m a n i f o l d o f M w i t h n o r m a l G - v e c t o r b u n d l e r , . v : v(lyftttt,M). D e f i n e v " a n d r ' r s i m i l a r l y . N o t e t h a t t ' r I M : t'^, and l'"'l N : \',\' S o m e t i m e s w e w i l l d e n o t e b y v M also the NH-normal bundle of MII in M. ('onsider the G-manifolds ltl : rtZlint Dr'r, N : l/\int Dr'' \-i Sr'r' and ll' :
W " , i n t D r , * . W e g e t a n i s o v a r i a n t h - c o b o r d i s m + W , tr t . N f s i n c c a n lr-ccrbordism is isovariant if and only if M a - W n and Nr - W,, are homotopy e q u i v a l e n c e s f o r e a c h H e I s o M ( H a u s c h i l d [ 5 ] ) . B y t h e i n d u c t i o n h y p o t h e s i s r f i , , 1 W ' . . \ i . , ü t e W h f . " ( , ü ) i s d e f i n e d a s \W,1r4,,V1 a n d h a s o n e o r b i t t y p c l e s s . [. e t
t l l l , , W H . , \ , f r r l W H . A / H / W H ) e W h t 1 , l Z n 7 W H l b e t h e W h i t e h e a d t o r s i o n o i t h c n o n e q u i v a r i a n t f t - c o b o r c l i s m ( t V r r / W H , M rrlWH, NH/WH)' The obvious mitp:
2 . 2 . . s : W h f l . ( M ) O Wh'1M n1WHl -, Whf1"1M)
i s a n i s s m o r p h i s m a s M ^ n M ^ i s a W K - h o m o t o p y e q u i v a l e n c e f o r a n 1 ' K e l s o . \ 1 . D c f i n e
2 . 3 . r f i . , 1 L ! ' . M , l { ) : ^ s ( r f ; . ( W , l v l . N t O lW riWH. Mrr,'wH. ,V'i'wH))
T l l E O R t T M 2 . 4 . ( E q u i v a r i a n t . s - c o b o r d i s m T h e o r e m ) ' . L e t M h e u G - r t r u r t i l ö l t l s u t ' l t t h i t l t l i n ü \ I rrl 7 5 .fitr t'ut'h H e Iso M.
( i ) T x ' 6 i s o t u r i u n t h - t ' g h o r d i s m s ( W , M , l , l \ a n d ( W ' , 1 \ 4 . N ' ) o t e r ! 1 7 u r e G - t l i l / e t r n r o r p h i t ' r e l M i l ' u n d o n l v ' i f ' t f t , , ( W , M , N ) r r r r , / r f l " ( W ' , M , N ' l u t l r e e '
1ii) 4rrt elentent in the isot'uriunt Whiteheutl ttroup Whfa,,(r\{\ t'tttt he reuli:t'rl rts rfa,, (14". I,/. rV) lrtr some isotariunt h-c'ohordisnr (W, M. ,N).
P r o t t f ' . S e e B r o w c l e r a n d Quinn [1], Hauschild [5]. Rothenberg [ 1 6 , ] l j A ( n o t n e c e s s a r i l y i s o v a r i a n t ) f t - c o b o r d i s m ( W . M . N ) d e f i n e s r ' ; ( W ' M , N - ) e W h t ' t ' \ / ) b i , . t l r e fo r m u l a : j * r u ( W , M , N ) : f 11: M - - W ) . B y T h e o r e m 2 . 4 , a n d t h e e q u i v a r i a t t t t r i a n g u l a t i o n t h e o r e m , t h e r e is a m a p ,
o : w h f l . ( M ) - w h G ( M )
u n i q u e l y d e t e r m i n e d b y t h e p r o p e r t y t h a t O ( r f . { . ( W , M , N ) ) : r';(W'.M,,\). for arty l r - c o b o r d i s m ( \ y ' , M , . N ) . D e f i n e th e d i r e c t s u m m a n d .
2 . 5 . W h t ( M l c - w h ( ; ( M ) to be the image of
O wh(zu (EwH X wH(,) Mo (")))
I j G i ' l - M l i -
u n d e r t h e i s o m o r p h i s m o f 1 . 6 , w h e r e I{ G I ? - - . M } l - i s t h e s u b s e t o f { G l ? -' M " l - represented by elements [r : GIH -- Xfwith Mnlx)a * O.In other words, we consider o n l y c o m p o n e n t s C o f M H c o n t a i n i n g a p o i n t x e C w i t h G " - H. For each xe M with G, - H, WH(x) acts freely on Mo(x), so @ sends the summand corresponding to Whl(MH(x)/WH(x)) to the summand of Whc(If) corresponding, via l'6, to
129
1 3 0
F ' R A N K C O N N O L L Y A N D W O L F G A N C I - Ü C ' T w h ( z n r ( E w H X wrr(.t)vtt('t)))' T h e r e f o r e ' o i s a m a p1 . 6 . < D : W h f l , l A 1 ) - w h t ( M ) .
F o r t h e r c s t o f t h r s p a p e r w e m a k e t h e f o l l o w i n g a s s u m p t i o n '
A S S U M P T I O N 2 . 7 . M h a s c o d i m e n s i o n 3 g a p s ' T h a t i s t o s a y ' d i m D - d i m C * | a n d 2 . w h e n ( ' e n n ( M * ) , D . n o ( M t l ) , c - D . H c K . M o r e o v e r , d i m ( A / " \ > 5 h o l d s fo r a n y H e I s o M .
THEOREM 2.8. (i) o is an isomorphism o.f aheliun (Jroups' (11) Ant' h-r'obordisnt oL:er ill is isr.rt'ru'ittttt.
P r \ o l ' . ( i ) T h e p r o o f is d o n e in d u c t i v e l y o v e r t h e n u m b e r o f o r b i t t y p e s ' C h o o s e ( H ) s o th a t H e I s o M i s m a x i m a r l . c o n s i d e r a n is o v a r i . n t rr - c o b o r d i s m ( t { 2 . M , N ) a n d d e f i n e ( W . \ f i , , { ) a s a b o v e . L e t
2 . g . r r f : w h ' 1 M , , i w H ) - w h t ( ^ 7 )
h c t h c c o m P o s i t i o n :
w h r ( M H / r y H ) - , w h w H ( M r ) ' w h N H ( M H 1 I ' ( s l v ) * ' w h N H ( s r ' ^ ' ) - - W h G ( G x *rSt'1a) - - Wht(AZ)
We claim that the following diagram commutes if k, r, and s are the obvious i s o m o r p h i s m s .
2.10. whfl"(ÄZ) o wh'1M,,7WH) - --whf..(M) fov - trt
L 0 f t _ i
I
l*" I
x * s M r ) ' 'Wnf tV) whf (.ü) o wh' (EwH
( f t is a n i s o m o r P h i s m b Y 1 . 6 ) .
B y d e f i n i t i o n , < D - s ( r G ( W , l \ 4 . r V t e t | 1 W , r 1 W H , M I I / W H , I V H / W H ) ) r s { ; ( W . M . N ) . T h e f o l l o w i n g c a l c u l a t i o n i n W h ! ( M ) i s a c o n s e q u e n c e o f 1 ' 2 ' T h e p h r a s e . i n Whfi{M)' means that all torsion elements are mapped to Whf(M) b y a h o m o - m o r p h i s m w h i c h i s o b v i o u s fr o m t h e c o n t e x t '
t r , t ( M c - W \ : r G ( M - i n t D r , , c - W - i n t D l s ' ) - r t t ( S t ' , c S v r )
+ r G ( D t , u c - D r ' r \ : f ( N [ - w \ - t r f ( r ' 1 M r r l w H ) c l 4 z r r i w H ) + r t ( M n 1 W H c W n l w H )
H e n c e . 2 . l 0 i s c o m m u t a t i v e . S i n c e < D e 7 i s a n i s o m o r p h i s m o f a b e l i a n g r o u p s b y i n d u c t i o n h y p o t h e s i s , t h e s a m e is t r u e f o r 0 - '
( i i ) N o t i c e th a t M x - MK is 2-connected f o r K e I s o M a n d ( w , M , N ) i s i s o v a r i a n t i f a n d o n l y if M K - - w x a n d N ^ - r w x a r e w e a k h o m o t o p y e q u i v a l e n c e s f o r K e l s o M ( s e e H a u s c h i l d [5]). The details of the induction o v e r t h e o r b i t t y p e s is l e f t t o t h e
r e a d e r . I
e'."r@-&,*.
T H E I N V O L U T I O N o N T H E E Q U I V A R I A N T W H I T E H E A D G R O U P Next we clefine maps * such that the following diagram commutes:
1 3 1
2 . r l . w
Namely, x sends rf,"(W. M, N)(resp. rG (W,M' N)) to j(M)* 'i(l{)* rf,'(w' lv ' M\ (resp' . i ( M \ * , j ( N ) * T o ( W , N , M ) ) ' H e r e j ( l { ) a n d T ( M ) d e n o t e t h e o b v i o u s i n c l u s i o n s ' T h i s i s well defined by Theorems 2.4 and 2'8.
W e w a n t t o e x p r e s s * o n W h f ( M ) i n t e r m s o f n o n e q u i v a r i a n t W h i t e h e a d g r o u p s a n d s h o w t h a t x i s a n i n v o l u t i o n o f a b e l i a n g r o u p s . A g a i n w e u s e in d u c t i o n o v e r t h e o r b i t types starting with the case where M has only one orbit type (H)' Let C be a component o f M lG : M alwH. Let rv,(c): z,(c) - - ,
{ t 1 } b e it s fi r s t S t i e f e l - w h i t n e y c l a s s a n d n ( c ) i t s d i m e n s i o n . E q u i p z n r(C) with the involution Lio' g - ' 2 ) " o ' w , ( c ) ( g ) ' q - | ' It i n d u c e s a n i n v o l u t i o n o n W h ( 2 , ( C ) ) M u l t i p l y i n g i t w i t h t h e s i g n ( - 1 ) n t c t w e g e t a n i n v o l u t i o n * ( c ) . T h e n t h e f o l l o w i n g d i a g r a m c o m m u t e s , w h e r e c r u n s o v e r n o ( M l G ) ( s e e M i l n o r [ 5 ] ) .
w h t ( M f tut
* , w h t ( M )
1."
-r--Wtrf."(M)
- | = l
- l
: l l
+ ^ / - \ v
O Wh(n' (C)) -91Q-- O Wh(n, (C))
T h i s f i n i s h e s t h e in i t i a l s t e p . I n t h e in d u c t i o n s t e p c h o o s e ( H ) , H e I s o M m a x i m a l ' N e x t w e p r o v e t h e c o m m u t a t i v i t y o f t h e d i a g r a m
-'---wht(M)
J l .
' 'wnf(M)
h ; ( M ) I
l * '
t hf;"(M)
w
2.t2. wht(Az) o w
whi(^Z) o w
This is a consequence of the following calculations in Whf tM):
rc(w.M, N) : f (l\4 U st'r Dt', c- l{z [Jsr'r Dt'*)
: t G ( N 4 - W ) - r c l S l ' r c S l ' 4 , ) + T G ( D t ' u c - D t r )
: t " ( w , M , , v ) - t r f ( r t ( w r r l w H . , M H l w H , N H / W H ) ) + rt ( w H lwH. M H lwH,Nrr/wH),
a n d . u s i n g 1 . 3 ,
2 . 1 3 . * f ( w . M , l u ' ) : r c ( l u ' c : w ) : r c ( / v - N U D v " D y w ) + r G ( N t W ) : r r ( w n , w H . N n r ' w H . N I H w H ) + t c ( ' \ - c w ' ) '
Hence *: Whf (M) -- wht(M) is an isomorphism of abelian groups. It remains to show that x is an involution. In the sequel, all torsion elements are understood to be mapped
h r ( M H l w H )
I
l [ - t r f * l
l l o - . I
h 1 ( M H l w H )
I
t
132
FRANK CONNOLLY AND WOLFGANG LÜCK into whf (M). Represent .x e whf (M) by ro( w, M,l{) and x (.x) by - rG (f,ry, M , f 1. then w e h a v e , b y d e f i n i t i o n ,x ( . r ) : r " ( w . N , M ) a n d * * ( , x ) - - r G ( W , N , M \ B l a s s u m p t i o n
' c G ( w
l ) w . l r l , l r / ) : ro(w,N,M) +- rG(fr,M, Ar) : *(.x) - x(.x) : Q.
Therefore rG(Wl) W,IV, lvl : g also, by Theorems 2.4 and 2.8. Therefore, r " ( f r ' . r l . , l f t + r G ( W . M , N ) : - x * ( . x ) * - x v a n i s h e s . T h i s f i n i s h e s t h e p r o o f t h a t t h e m a p s x i n 2 . 1 1 a r e i n v o l u t i o n s o f a b e l i a n g r o u p s .
S i n c e th e r e is a n a l g e b r a i c d e s c r i p t i o n o f t r f i n L ü c k [ 3], we obtain, all in all. an a l g e b r a i c d e s c r i p t i o n o f x : W h f ( n f ; - - W h t ( M ) .
W e c l o s e th i s s e c t i o n b y c o l l e c t i n g s o m e e l e m e n t a r y p r o p e r t i e s o f x a n d p ( S ( O ) * . L F . M M A 2 . 1 4 . L e t i : i M - , M h e t h e in c l u s i o n o f ' t h e h o u n t l a r l ' o . / ' t h e G - m u n i J b l d M . T l t e n n ' e h u t , e * i x - i* *
P r o o l ' . L e t ( W , ? M , L ) b e a f t - c o b o r d i s m o n ? M . l d e n t i f y i M x I c M : t t ' t x I l ) w i t h a c o l l a r . U p t o s t r a i g h t e n i n g t h e a n g l e , w e h a v e a n f t - c o b o r d i s m ( V , M , N ) w h e r e
I . i s t f x l f l r ' , r W x I , M i s M r ' , 0 ) ' a n d l V i s i V - i n I M . W e w a n t t o c o m p u t e r t ; ( N , : v ' \ i n W h o ( , V ) . L e t n 7 b e M - i n t ( ? M x I \ .
i ' ; i , v c - l " ) : " c G ( L t 1 U r , , ; , w x i 1 f ) , . , r ' , , ( h ' l U a M x 1 ) c w x I U , ' n r ' , M x I ) : ' c G ( L r 1 U , . . , , w ' x f 1 c - w x I )
: r G ( L x 1 c w x l l - r G ( L x i l c w x i l l + r t t ( w x i l c - w x i l ) : ' t " ( L - w l - 2 . { ; ( L c , w l : - - t t t ( L c w ) .
W c a l s o h a v e in W h G { M ) : r ( ; ( . M c - v ) : t ' o ( M -
: rrr(iM x : r u ( ? M x
W x l l ) ; u , . t M )
/ l i v " , M c W x 1 ! i , u , , M ) I c W x l ) : f ( ? M c W ) .
aII
altl x I
T H E I N V O L U T I O N O N T H E E Q U I V A R I A N T W H I T E H E A D G R O U P Hencc. we gct
x i * ( r G ( v v , ? M , L ) ) : * ( r G ( v , M , N ) ) : r G ( v , N , M ) - - i * t G l L - w 1
- i* * (ro (w, eM, L)). Il
L E M M A 2 . 1 5 . L e t ( ! M b e a G - O ( n ) - u e c t o r b u n d l e o n a c ' k t s e d G - m a n d ' t t l d M , a n d le t i: Si - D( he the inclusion' The we haue
( i ) * p ( D O * : p ( D C ) * * - i * P ( S { ) * * , ( i i ) * p ( S O * : P(SOt x,
( i i i ) i * * : - * i x .
p r o g f ' . ( i ) c o n s i d e r t h e h - c o b o r d i s m ( 1 4 l , M , l V ) o v e r M . L e t 4 l w b e a G - O ( n ) - v e c t o r b u n d l e w i t h r 7 l , : l . T h e n ( D 4 , D ( , D q f i u s r y ) i s a n h - c o b o r d i s m o v e r D ( , a n d w e h a v e
f 1 D 4 , D 1 , D 4 1 N u S 4 ) : P ( D i - ) * r G ( W , M , N ) ' W e g e t i n W h G ( O O :
* p ( D O * ro ( W , M , N ) : r G ( D 4 . D q 6 u S 4 , D o
: rG(Dq 1,, - Dry) - ro(S4 lrv - Sry) : p ( D C ) * r G ( N c W ) - p ( S O * r G ( N c ' w )
: p ( D O * ( x f ( W , M , . N ) ) - p ( s o * ( * r G ( w , M ' N ) ) ,
which proves (i). Property (ii) is verified similarly, and (iii) follows from Lemma 2'14
tr
3. Maps between G-Manifolds
Let ( /, ?.f \: (M. a M\ -- (,ry, eA/) be a G-homotopy equivalence of pairs of G-manifolds' We define a homomorPhism
3 . 1 . < D r : U G ( l { ) - - , w h f ( l " l )
as follows. Any base element ly:GlH -- lvl in uo(N) can be represented by /' .r: GIH --+ N for some x: GIH --' M' Let <P: tPv ''-l'*tpN be the OrG- e q u i v a l e n c e u n i q u e l y d e t e r m i n e d b y D E G ( I q - t ) : I (see Lück tl2l). From rp, we get f o r a n y x e M H a n d H c G , a n H - h o m o t o p y e q u i v a l e n c e
q(G lH\(,x)",, : T M', -- Tl{i'
between the one point compactifications of the tangent spaces' In the sequel' only input we needfrom rp is the desuspension of q(GlH)(x)"" denoted by g' : sT M ' STN... Recall that the Burnside ring A(H)acts on Whrr(STN.,). Let Or(["vj) denote
r 3 3
the the
F R A N K C O N N O L L Y A N D W O L F G A N G L Ü C K
1 3 4 image of
1 1 - e H ( s r N r ) ) ' r H k p * ) e W h l r ( s r ' r y , ) ' under the comPosition
w h ' ( s r N , ) - 4 " w h " ( { * } )
i n d ' w h o ( G l H ) } ' * 'whG(N)' One easily verifies that oy([y])e whf (N)'
2.j.
Recall that all G-manifolds are supposed to satisfy Assumptton
L e t r o u , a i l e w h c ( 1 t r ) b e f ( f ) ' - , * ' o Q . D a n d l e t l o ( N ' f l { ) e u o ( N ) d e n o t e lG(.N) - i*Xu1fN), where i: öN --+ l{ is the inclusion map' It is easy to verify that I s o M : I s o N a n d t h a t f i n d u c e s a n i s o m o r p h i s m , , o ( M r ' ) : n o ( N H ) fo r e a c h H i n Iso M. It follows that rG(f,Af ) and ro(f) lie in whi('ry]' The involution * on whf;(N) was introduced in the last section'
T H E O R E M 3 . 2 . L e t ( f , a . f ) : ( M , i / M ) - ' ( N , eN) be a G-homotopJ' e q u i u a l e n c ' e o f ' p a i r s o l G - m a n i f u l d s . T h e n , x o ( f ) : - * r o U , A f \ - * Q . . ( x o ( / v ' a ^ i ) ) '
proo.f. By Kawakubo [8], we can assume that (M, aM)and (N' crN) are embedded in (DV.SV) fora G-representation Iz with normal bundles t't and r'" and the following is t r u e .
3.3. (i) There is a pair of G-fibre homotopy equivalences (f, Sf): (Dv u'S''r) --+ (Dvry' $1'") covering f,
( i i ) T h e i e i s a n e m b e d d i n g ' b : (D v ' ' S r ' t ) - - + ( D t ' 1 " D ' ' " - i n t j D v " ) s u c h t h a t t h e G . m a p s ( D v " , $ 1 , ' ) - + ( D v " , D v " - int }Dv") induced by p and b are G h o m o t o p i c ' M o " o u l t ' j D ' ' " c h ( D t ' ' ) h o l d s ' T h e h o m o t o p y s e n d s D t r I ? M t o D v * il N '
( i i i ) T h e i n c l u s i o n S ( v " ) H c D ( v ^ , ) H i s 2 - c o n n e c t e d f o r a l l H , - G . Moreover, St'^' and Dr'^, satisfy Assumption 2"7 '
To achieve (iii), one may have to enlarge I/' Consider the following cobordism ( W . jS r , , , X ) g i v e n b y ( h ( D t , . ) - i n t I D v " , 1 S r " , b ( S r ' r ) u h ( D r ' - | iM ) - i n t 1 ( D t ' n I i' N ) ) S i n c e b ( D v r l c D r , " i s a G - h o m o t o p y e q u i v a l e n c e a n d w c - h ( D t ' ' ) a n d Dr,"-int }Dr," c Dr,, induce 2-connected maps on the H-fixed point sets for any H c G' the inclus ion W c Dv"-int lDu" is a G-homotopy equivalengt by excision' Moreover' b e c a u s e h ' . D r ' r - - D ' ' f i s G - h o m o t o p i c r o []' we get' in Whf(Dr''):
34 '"'\_:"rloä;::';:,
: T G ( h : D v , . - I ) v 1 , ' ) : f ( ß . . D , , * -- D t , N ; : p ( D r ' 1 ' ; x t G ( / ) '
Since }S," c Dv*-int }Dv" is a simple G-homotopy equivalence, }S''" c- W is a G homotopy equivalence, and in wttf totr) we obtain, by means of 3'4:
3 . 5 . r c l j s v " c W ) : - P(DI'N ) * r G ( f ) '
t
T H E IN V O L U T I O N O N T H E E Q U I V A R I A N T W H I T E H E A D G R O U P I 3 5 N o w h : S r , ^ , - - + b ( S l n a ) i s a s i m p l e G - h o r n o t o p y e q u i v a l e n c e a n d i t s c o m p o s i t i o n w i t h b ( S l n r ) c D r , ^ * - i n t l D r , , i s G - h o m o t o p y e q u i v a l e n t t o S B : S v r - - D t ' " - i n t l D v ' . H e n c e b ( S r , . , ) c - D r , - - i n t jD r ' " i s a G - h o m o t o p y e q u i v a l e n c e . B y T h e o r e m 1 . 1 3 , w h e r e O . , w a s d e f r n e d . w e g e t . in W h c l S r ' " ) :
3 . 6 . r G ( b ( S l n a ) c D r , r - i n t l D v r ) : r G ( S l J : S v r - - ' S t ' N ) : p(SvN)* r G ( . f | + O s 1 l ( / ( ; ( l { ) ) ' L e t . i : S r , , - - D y n d e n o t e in c l u s i o n . B e c a u s e o f 3 . 4 a n d 3 . 6 , w e o b t a i n i n W h ! ( D r ' " ) :
3 . 1 r G l b l S r ' r ) c : W ) : r o ( b ( s v r ) - D v r - i n t l D v " ) - rG(l,l/ c. Dv,,1 - intlDur)
: i*p(svn,)*f (f) * /*orp(xo(N)) - p(DvN)* r o ( f ) ' S i m i l a r l y , w e g e t i n W h f ( D v " ) :
3 . 8 . r c l b l s r ' , I A M ) c b ( D v * l A M - i n t l D v " l d N )
: i*p(Sv, )* i*rG Qf) * i*orp(,*lc(AN)) - p(Dr'" ;*i* rc(ff).
C o m b i n i n g 3 . 1 a n d 3 . 8 , w e o b t a i n , in W h f ( D v r ) :
3 . 9 . r G ( x - w ) : r o ( b ( s v r ) - w ) - r c l b l s v r Ii'M) c b(Dvrli,M) - i n t ]D v , l i N )
: /*p(Svrs )* to (J, eJ ) +/*@rp(to(^/,olN)) - p(Dv il* rG (.f. t.l'\- Let lr : X .- Sy, be the obvious homotopy equivalence, and now identify lSt'" with Sr'^-.
B e c a u s e o f L e m m a 2 . 1 5 , f o r t h e t o r s i o n o f ( W , ) S v r , X ) i n W h f ( O v r ) w e o b t a i n : 3 . 1 0 . / * r o ( W , l S v r , X ) : i * * k * r G ( W , X , j S v r ) - - * i x k * t o ( W , X , ä S t ' " ) ' W e n c r w c o n c l u d e f r o m L e m m a s 2 . 1 5 . 3 . 5 , 3 . 9 , a n d 3 . 1 0 , t h a t i n W h f ( D r ' r ) :
3 1r ':::.;:,,,;l],,-
rG(1,?f)- xj*o,p(ro(N,..N)) 1 *p(Dr,. )*f U.ü'\
: /*p(Sy")* * rn(./, ?.f) - *-1*<D5p(XG(N, i N ) ) * p ( D r ' ^ , ) * * r G l . f . i ' l ' l - /*p(Sv,v)x x tG (.1', ot.f)
: p(Dr',)* * rc (.f. A . f ) - x 7 * O r r ( / G ( N . f N ) ) .
H e n c e w e g e t , b y a p p l y i n g p ( D v o ) * t o 3 . 1 1 , a n d u s i n g P r o p o s i t i o n 1 . 7 , th a t , i n
whf (Dr'r):
3 . 1 2 . ro ( . f ) : - * r G ( f , i . J ) + p ( D v r o ) * * i * O r r ( x o ( l t , a N ) ) .
Therefore it suffices to verify
3 . 1 3 . * @ . r : - p ( D v r o ) * * i * O r p : U c ( r u ) -- ' W h t ( N ) . For this we need the following lemma.
I
r36
F R A N K C O N N O L L Y A N D W O L F G A N G L U C K L E M M A 3 . 1 4 . L e t .l': SV' -- SV and g: SW' --, SWbe G-homotopy e q u i t a l e n ( ' e s b e t v ' e e ns p h e r e s o . f ' G - r e p r e s e n t a t i o n s . T h e n w e h a u e in W h c ( [ * ] ) :
( 1 - z,'(SV x SW))-rG(f* s) : (1 - zo(SV)).rG(.f\ + ( l - x G 6 W ) ) . r G k t ) P r o o f ' . T h e j o i n X * Y i s d e f i n e d a s C o n e ( X ) x Y U r . r X x C o n e ( Y ) . N o w t h e
result follows from 1.2 and 1.4 I
T o v e r i f y 3 . 1 3 w e h a v e to s h o w : 3 . 1 5 . O / ( ) ' ) : - * p ( D v . " ) * * / * 0 s p 0 )
f o r a n y y ' : G l H - - l { o f t h e f o r m y : f " x w h e r e x : G l H - - + M i s a G - m a p .
Since * commutes with codimension zero embeddings, we can replace l{ and Dr'^- by DT and DT x DW, where ?. and W denote the fibers at y of Tlü and ll{. Since
* c o m m u t e s w i t h i n d f i : W h r r ( X ) -- W h G ( G x n X ) , w e c a n a l s o a s s u m e H : G . T o e s t a b l i s h 3 . 1 5 , w e h a v e t o p r o v e
3 . 1 6 . ( l - e G ( S T ) ) / * r G ( E , ) : - * p r * r * r G 1 S B , )
w h e r e r : S W - - + D T x D W a n d r ' : S I - - D T x D W a r e i n c l u s i o n s a n d p , : D T x D W - - DT is projection. In view of Lemma 3.14, this reduces to proving
3 . 1 1 . r * t o ( e ) : * p r * ( 1 - e G 1 S W 1 7 t ' * r G ( 8 , )
because the join of SB, and E, is homotopic to the identity.
B u t a c c o r d i n g t o 1 . 8 a n d 1 . 9 , t h e m a p
( x D W - x SW): WhG(Df) -' whG(DT x DW)
sends any element pr*(a) to (t - eG(SW))a, and the map x DW: WhG(Dl; --' W h c ( D T x D W ) i s i n v e r s e t o p r *
So to prove 3.l7,it suffices to show that the diagram below commutes:
whf (Dr) * ,whf (Dr)
l l
I t , o w - x s l f ) l x D w
J v
whf(D r x DW) -\whf @T x DW)
But this is clear from 2.15(i).
This completes the proof of 3.2.
4. Examples and Applications
We begin with some illustrations of the results of Section 2 by computing the involution on Whf(N) in the case of a semi-free action. Namely, let N be a G-manifold such that G and { 1} are the only isotropy groups. For simplicity, we assume that N and Nc are connected. Assume n : dim(Nc) and n * k: dim(N). Assumption 2.7 reduces to: n2 5 and k>- 3. Write rc: rr.(lüG) and | : r-r(lü). Since lü has a fixed point, G
T
t
T H E IN V O L L J T I O N O N T H E E Q T J I V A R I A N T W H I T E H E A D G R O U P 1 3 ] acts on I- ancl we can consider the semi-direct product I- X, G. One easily verifies I - * , G : n t ( E G x c l{). Let w,(No) " n - - +
[ t l ] b e t h e 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 , v ( ; . w e e q u i p n w i t h t h e x , , 1 N G ) - t w i s t e d i n v o l u t i o n . r i o ' 9 1 - ' 2 ) n ' [ ' r ( N G )( t l ) ' l l - t ' L c t x : w h ( n ) - - + w h ( n ) b e t h e i n d u c e d in v o l u t i o n m u l t i p l i e d w i t h t h e s i g n ( - 1 ) ' ' D c f i n e * : W h ( l - t . , G ) - W h ( l - x , G ) a n a l o g o u s l y u s i n g t r ' r ( y V - l v r c , ' G ) a n d ( - 1 ) ' t ' ' Consider the normal G-vector bundle 1' - l'(NG, N) of l{c in N and the induced f i b r e b u n c l l e p : S v i G - , N G . N o t i c e t h a t n , ( S v / G ) : r ( x G ' I n L ü c k [ 9 ] ' t h e t r a n s f e r p * : W h ( n ) - - . ' W h ( n x G ) is d e f i n e d a l g e b r a i c a l l y . T h e o b v i o u s m a p i : n x G - ' I - x . G i n < J u c e s i * : W h ( z x G ) - - + W h ( l - x , G ) . T h e n t h e f o l l o w i n g d i a g r a m c o m m u t e s b y t h e r e s u l t s o f S e c t i o n 2 .
w h i ( N ) - w h ( z ) o w h ( f x , G )
I
t *l [ . o l
i l II l L - i * 1 ' * * * - l
+ *
w h : i ( N )
Let V be rhe normal G-slice of tVG in N. Then p has SVIG as typical fibre'
The algebraic transfer depends only on the pointed transport of the pointed fibre o ( p ) : n x G - l s v l c , s v l c ) * , i . e . a h o m o m o r p h i s m i n t o t h e m o n o i d o f p o i n t e d h o m o t o p y c l a s s e s o f p o i n t e d s e l f - m a p s o f S V I G . N o w s u p p o s e t h a t G h a s o d d o r d e r ' T h e n a n y s e l f - G - h o m o t o p y e q u i v a l e n c e s v - - , s z i s G - h o m o t o p i c to t h e i d e n t i t y a s V G : 0 a n d A ( G ) * : I t l ] h o l d s . I f q : N G x V - - + N G i s t h e t r i v i a l G - R k - v e c t o r b u n d l e , t h e n o ( p ) : o ( q \ a n d , h e n c e , t h e t r a n s f e r m a p s p * a n d q * a g r e e ' B y t h e p r o d u c t f o r m u l a r 7 * a n d p * v a n i s h , a s 1 ( s v l G \ i s z e f o . H e n c e ( - i * P * ) x i s t r i v i a l a n d t h e i n v o l u t i o n o n W h f ( l { ) i s g i v e n b y t h e d i r e c t s u m o f t h e i n v o l u t i o n s o n W h ( n ) a n d W h ( f x , G ) d e s c r i b e d a b o v e (c o m p a r e w i t h T h e o r e m 4 ' 2 ) '
N o w s u p p o s e t h a t G i s Z l 2 V . T h e n t w o p o i n t e d h o m o t o p y e q u i v a l e n c e s l ' a n d u "
S V i G - - S V l G a r e p o i n t e d h o m o t o p i c , i f a n d o n l y i f d e g ( / - ) : deg(u- \e I t 1 i holds for the lifts I -
and (l- . Therefore we can interpret o(p) as a homomorphism n x G - - { + 1 } . L e t w , ( s r , ) e H l ( N c ; z l 2 z ) b e t h e 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 S ( r , ) I N G . I f w e w r i t e G : { + l } t h e n o ( p ) s e n d s ( t ' , r 7 ) e f t x G t o n ' 1 ( S r ' ) ( r ' ) ' 4 ' N o w c o n s i d e r p * p * : W h ( n ) - - W h ( n ) . L e t S w ( z ) b e t h e G r o t h e n d i e c k - g r o u p o f V n - m o d u l e s w h i c h a r e f i n i t e l y g e n e r a t e d f r e e o v e r Z - l t a c t s o n W h ( z ) b y 8 , ' ' l f Z ' s t a n d s f o r Z ' equipped with the n-action coming from n" (sr') and z is the trivial zn-module then p * p * i s m u l t i p l i c a t i o n w i t h l z ) + a - \ u l z " ' ) e s w ( n ) ( s e e L ü c k tl0l) Hence' PxP* and p * a r e n o t t r i v i a l in g e n e r a l . I f k i s e v e n a n d w r (Sr') : 0 p* P* is multiplication b y 2 ' E v e n i f k i s o d d a n d x 6 v l G \ : / ( s v ) : 0 . P * P * a n d p * c a n b e n o n - t r i v i a l f o r a p p r o p r i a t e n a n d r t ' t ( S r ' ) .
o n e c a n a l s o g i v e e x a m p l e s o f a g r o u p G a n d a G - m a n i f o l d N s u c h th a t N h a s tw o o r b i t t y p e s , a l l f i x e d p o i n t s e t s a r e e m p t y o r s i m p l y c o n n e c t e d ' a n d t h e i n v o l u t i o n o n W f r t t r u l i s n o t t h e d i r e c t s u m o f i n v o l u t i o n s o n t h e s u m m a n d s ' H o w e v e r ' in s o m e f a v o r a b l e c a s e s t h e in v o l u t i o n o n w h t ( N ) s p l i t s i n s u c h a s i m p l e f a s h i o n . N a m e l y , m a k e t h e fo l l o w i n g a s s u m p t i o n s . L e t t h e o r d e r o f G b e o d d . C o n s i d e r a c o n n e c t e d G - m a n i f o l d
Lr.--*- -.,. - -...- _ .- _._.- . *,*-*"--i-
1 3 8
F R A N KlV such that for any .x: GIH --+ l/ there is an r e s l H t ' t 1 V ) a n d t h e n o r m a l H-slice vlNH(.x), N ) , c o n d i t i o n is always fullfilled for abelian G.) Since
| ---, n,(N"(-r)) -:L--'n,(EWH(-x) X *"r,r ru"(r)) - wH(,x) --, I
i s e x a c t and wH(.x) has odd order, there is a homomorphism t,(-x): n , ( E w H ( . x ) X wH(,) l { " ( . t ) ) - - lt 1} uniquely determined by the property r,(-x)i* - },r,1(l/)llv'(,x).
Let
* : W h ( n , ( E W H ( - ' ) x *ur,r lV,t(,r))) - - Wh(n,(EWH(,x) X wu,,., l V r r ( . r ) ) ) b e t h e r ' ( . r ) - t w i s t e d i n v o l u t i o n multiplied with the sign (- l)dimrrt.
T H E O R E M 4.2- With the assumptions a b o t , e , t h e fiilktv,intl diuurum contnlutes v ' l r e r e t h t ' .s t t t t t r u n s () r e r lt,Gl? - lVl \
wht(t/) =,
O wh(n, (EWH(-x) X *rr_., lg"(-r)))
l * l r n *
J W
w h p c ( N ) = ,
@ W h ( z , ( E w H ( x ) X w H ( , ) l r " ( - * ) ) ) .
P r o o f . T h i s follows from 2.13 and Theorem 1.12. f l
T h i s s p l i t t i n g result is quite helpful in the calcularion of H*(Zl2Z:WhEp(M)) in C o n n o l l y and Kozniewski [2], where crystallographic m a n i f o l d s c o r r e s p o n d i n g t o c r y s t a l l o g r a p h i c g r o u p s I - with holonomy group G of odd order are examined.
T H E O R E M 4 - 3 . C o n s i t l e r a G - h o n t o t o p t ' e q u i t t u l e n t ' e ( 1 . i 1 1 : ( M , i M ) - - ( l / . i l / ) h e t x ' e e n G - m u n i / b l t l s . S u p p o s e n o Q H ) : z n ( f N I r ) - z o ( l / I I ) and Tr(itl ..x): n,(i,Ä{r..xy - * t r r ( l { I I . x) ure hi.jet'tire.fir u n y ' H c - G urtd r e f N H . A s s u n t e o r y u r r o o / t l t e /ollotrinq t'onditiorts;
( i ) T h e n t u p < D , : U G I N ) -- Wh,;( N l u p p e a r i n g i n 3 . 1 is z e r o .
1 i i ) I . / q : t p u - r . l ' * t p * t l e n o t e s t h e u n i q u e O r G - e q u i t , u l e r y e l , i r l r D E G ( l . d : l , t l e r r
<p(G,'H)(x),'1,: TN'li -- rl/i. is u sintple H honrotopt' equit'ulent'e för unt, H c (i untl re I/III .
( i i i ) F r r r - r e MIt the G.-representutions T M , u n d TN s, ure litteurlt,G..-i.sr.ututrpltit' untl G is tlrc protlut't o.f'u tlroup o/'odd order und u 2-uroup.
( i r ' ) / . " ( N ) e U t i l / V 1 , , r / r , l N , l N ) e U , t ( N ) t u n i s l t e s .
T h e n : I f ' o n e o / t h e e l e n t e n t s , r G ( . f ) e W h c ( N ) o r r G ( f - , i / ' ) e W h c ( N ) r . n n i s h e s . t h e n u l l r l t e e l e n t e i l s r G ( a . l ' ) e w h c l r l u ) , to U')e whclN ), und rc U, il le whGlN l ure zerr.
P r o o / ' . I f ( i ) o r ( i v ) i s t r u e , t h i s f o l l o w s from the formula tG(il - -*r,r(. / , i f ) -
* o , { / G ( l { . ( ' l / ) ) o f T h e o r e m 3.2. Norice rhat i*:whclllv) -- whlt(l/) is bijective by a s s u m p t i o n a n d 1 . 6 . O b v i o u s l y , ( i i ) i m p l i e s (i ) M o r e o v e r , u n d e r condition (iii), any G homotopy equivalence T Mi -- rl{:r.. is G homotopic to a G map inducecl by a linear G - i s o m o r p h i s m a n d . h e n c e . i s s i m p l e . T h i s f o l l o w s from a result of Tornhave I I g]. The p r o o f is carried out in detail in Dovermann and Rothenberg [ 4 ] . t r
C O N N o L L Y A N D w o L F G A N c I-I]cx N H ( , r ) - r e p r e s e n t a t i o n / s u c h t h a t a r e H - h o m o t o p y e q u i v a l e n t . ( T h i s
t
T H E I N V O L U T I O N O N T H E E Q U I V A R I A N T W H I T E H E A D G R O U P
This theorem is an important tool for the proof of the equivariant n-z-theorem for G-manifolds and simple G-homotopy equivalence (see Dovermann and Rothenberg [ 4 ] , L ü c k a n d M a d s e n [l 4 ] ) .
Finally, we want to illustrate by an example the appearance of the correction term Ort/c{lri, f l{)) in the formula of Theorem 3.2. Notice that it does not appear in the nonequivariant case. Namely, consider a G-homotopy equivalence öf: SV -' SW b e t w e e n s p h e r e s o f G - r e p r e s e n t a t i o n s . D e f i n e U,Afl:@V,SV)--.(DW,SW) by coning.
Suppose for simplicity that SVG is nonempty. Then Llo(DtZ) is just A(G) and r p . . : S T ( D V ) , - - S T ( D W ) r , i s g i v e n b y r e s f i ( d / : S V - - - , S i l 2 ) f o r H c G a n d . x e M H . M o r e o v e r , 0 r i A ( G ) - - W i n G . J W ) s e n d s th e b a s e e l e m e n t lC l H ) t o i n d f i r e s f i ( l - X G 6 W ) ) ' T G G . D .
Hence we have
4 . 4 . Q r ( x o ( D W , S W ) ) : x G I D W , S W ) ' ( 1 - x o ( S W D ' r c ( A f ) : r o ( A . i l . Now, from Theorem 3.2. we get
4.5. rG (.f ) : - * ro (.f, Af ) - *Q y(xo (DW, SW)) O b v i o u s l y , rG ( . f ) is z e r o . H e n c e , 4 . 5 re d u c e s t o
4 . 6 . 0 : * r o Q . f ) - * @ s ( x o ( D W , S W ) )
B u t 4 . 4 a n d 4 . 6 m a t c h u p . S o t u ( . f ) I - * r o ( f , e . i l i n g e n e r a l .
Acknowledgement
The first author wishes to thank the Sonderforschungsbereich in Göttingen for its hospitality and support during the stay in August 1987 when this paper was written.
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r 3 9
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