manuscr ip ta math. 60, 387 - 395 (1988) manuscripta mathematica @ Springer-Verlag 1988

SPHERICAL RESOLUTIONS FOR COMPACT LIE GROUPS

STEFAN BAUER AND ALBERT SCHNEIDER

Let G be a compact Lie group and A a G-space. When does there

exist a relative G-CW-complex (X,A) with free G-action on X\A , such

that X has the homology of a sphere? This paper gives sufficient con-

ditions, which can be used for the construction of homotopy represen-

tations.

i. NOTATION AND STATEMENT OF RESULTS

Let G denote the identity component of G and G a subgroup of G o p

containing G o , such that Gp/G ~ = Fp is a p-Sylow subgroup of G/~=F.

The dimension g of G is assumed to be nonzero. All G-spaces are

assumed to be finite-dimensional and of finite orbit type.

Homology modules will have coefficients in a fixed subring T c Q of

the rationals. Let M(p) denote localization of an abelian group M at

the prime p . A relative G-CW-complex (X,A) is called a (T,n)-reso-

lution of A, if

i) X\A has free G-action and dim X\A ~ n

ii) X is simply connected, H,(X) = 0 for * % [n-g,n] and Hn(X)

is T-free.

A (T-n)-resolution (X,A) is called spherical, if H,(X) ~ H,(sn).

Note that H,(X) is a graded left module over the Pontrjaginring

R = H,(G) .

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BAUER-SCHNEIDER

Before we state the main theorem, we need further notation: A group H

is called p-toral, if H = H and H is a torus. A subgroup H < G p o

is called sub-p-toral in G, if there is a p-toral subgroup of G con-

taining H .

(i.i) THEOREM: There exists a spherical (T,n)-resolution of A , if

for each prime p ~ T x there is a G -complex Y(p) and a G -repre- P P

sentation sphere S(p), such for sub-p-toral K < G the following P

conditions hold:

i) H,(A) is finitely generated, H,(A) = 0 for * $ n 3 3+g and

Hn_I(A) is T - f r e e .

~H n for K = i

H..~(y(p)K)(p) ~ ,(S * S(p))(p) K(S n(K)* with n(K) ~Z ii) ~ IH,((A * S(p~K~p) ~ * S(p)K)(p )

otherwise

iii) The isomorphism in ii) respects the WGpK = NGpK/K-action, if

K is maximal.

(1.2) REMARKS:

Up to conjugation a maximal p-toral subgroup is unique. An appro-

priate name would be p-toral Sylowgroup. To see the uniqueness, let N

be the normalizer of maximal torus in G . The Euler characteristic

x(G/Np) equals IN/Npl ~ 0 mod p . If K is p-toral in G , we know by

Smith theory x(G/N~) ~ x(G/Np)mod p , so in particular G/N~ _ ~ ~. This

shows that K is subconjugate to Np .

If K is p-toral, A K has to be a Z(p)-homoiogy sphere by Smith

theory. So in condition ii) we postulate that Smith theory also holds

for sub-p-toral subgroups.

Condition ii) actually is a condition on the dimension function n,

which assigns an integer to each sub-p-toral K : There has to be a ~p)-

homology sphere Y(p) with dimension function, say m , also defined

on sub-p-toral groups in G , such that m and n differ by the P

dimension function of a real representation.

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BAUER-SCHNEIDER

In [i] this theorem is used to construct homotopy representations: It

is shown that the dimension function of a homotopy representation equals

the dimension function of an element of the real representation ring

RO(G ) on the sub-p-toral groups in G . This implies that condition ii) P P

holds. Condition iii) follows from an Artin-relation [3].

It is a well-known fact that a graded R-module M with M, = 0 for

, ~ [n-g,n] is projective iff Mn_g is a projective_ TF=Ho(G)-module

and multiplication gives an isomorphism R ~ | --~ M with R ~

We deal with such projectives only,

This paper makes use of the following two results:

(1.3) THEOREM (Schneider, [6]): Let n ~ 3+g , let Hn_I(A) be a free

T-module, Hn(A) = 0 . Let (X,A) be a (T,n)-resolution, H,(X) ~ M|

with F R-projective. Then there is another (T,n)-resolution (Y,A)

and a map (Y,A) --~ (X,A) , extending id A and identifying H,(Y) with

a R-submodule of H,(X) isomorphic to M .

(1.4) THEOREM (Oliver [4]): Let X be a G-CW-complex with H,(X) = 0

,, (X) a free T-module. If for all p ~ T x for -~ ~ [n-g,n] and with i n

and all p-toral subgroups H ~ I of G the fixed point set X H is

Z(p)-acyclic, then H,(X) is a projective R-module.

This paper contains a proof of theorem (i.i), a result of joint work

while both authors were working on their theses [i] and [6] . We are

grateful to our advisor Professor tom Dieck for his support.

2. PREPARATION OF THE PROOF.

The idea of the proof is to construct a nice "G -map" Y(p) -, X, S(p) P

and so analyse H,(X) . But first we have to beautify Y(p) and X .

(2.1) LEMMA: Let H be a subgroup of G which fits into an exact se-

quence H I >--, H -->> H 2 , such that H I is sub-p-toral in G and H 2

is sub-p-toral in WH 1 = NHI/H 1 . Then H is sub-p-toral in G .

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BAUER-SCHNEIDER

PROOF: Let X be a G-space and K be sub-p-toral in G . Choose a

p-toral P < G containing K and let K be the kernel of

K ~ P -->> P/P . Bu Smith theory we have the mod p congruences

x(X) ~ x(X P~ ~Px(x ~) ~ x(X K) , as Po and Po/K are tori and K/K

is a p-group. As in remark (1.2 i)) we have therefore

H H x(G/Np) -- x(G/N i) _ x((G/N p i) 2) = • 0

for a maximal p-toral subgroup

to N �9 P

N of G . Thus H is subconjugate P

Let Iso(Z) denote the set of isotropy groups of a G-space Z .

(2.2) LEMMA: For each p there exists a G -CW complex P

G -representation sphere SW(p) , together with a G -map P P

Y(p) * SW(p) , so that the following conditions hold:

A(p) and a

f : A(p) --,

i) There are only sub-p-toral subgroups in Iso(A(p))

ii) If H,K ~ Iso(A(p)) , then H n K ~ Iso(A(p))

iii) H,(fH)(p) is an isomorphism for sub-p-toral H < G

iv) If H,(Y(p) H * SW(p)H)(p) = 0 for * ~ k , then the topological

dimension dim A(p) H is less than k .

REMARK: Condition iii) allows us to replace Y(p) and S(p) by A(p)

and S(p) * SW(p) . The other conditions are needed for our "nice map".

(2.3) LEMMA Let V be an orthogonal G -representation such that Iso(SV) P

contains Iso(X * S(p)) , Iso(Y(p)) and the p-toral Sylow subgroups of

G . There exists a multiple W = I.V with the following properties P

i) if K,H ~ Iso(SW) , then also H n K ~ Iso(SW) .

ii) SW H and hence (Z * SW) H are either empty or (3+g)-connected

for H < Gp and any Gp-space Z .

iii) Iso(SW) consists of finitely many conjugacy classes.

iv) dim Z H ~ dim Z K for K~H~ IsoSW and Z ~ {X*S(p)*SW;Y(p)*SW}.

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REMARK: We will construct A(p) with IsoA(p)=~={H ~IsoSWiH is sub-p-toral}.

PROOF OF (2.3):

i)

Take i ~ 5+g + dimX + dimY(p) + dimS(p) .

and k of W with isotropy groups H and K Elements h

contained in wKN K. If L ~ KNH , then W L cannot contain both

and k . Therefore dim W HNK ~ dim W L and hence

dim W H N K ~ dim wL+g ~ dim( U W (L)) .

L~HNK

So H N K has to be an isotropy group of W .

are

h

ii) and iv) are trivial; iii) holds because the compact manifold SW

has finite orbit type. �9

(2.4) LEMMA: For 0 + q ~ Hg_I(G) there exists a $ ~ HI(G) with

q " ( ~ 0 .

PROOF OF (2.4): We can assume G is connected. Obviously the lemma

holds if G is simply connected or if G is a torus, as in the latter

case H,(G) is an exterior algebra with generators in HI(G) . Hence

the lemma holds in case of a product of a simply connected group and a

torus. In general G is a quotient G/K of such a group G by a finite

group K . Because of the transfer IKl.q lies in the image of

H,G --, H,G . The claim now follows, since Hg_I(G) is torsion free.

PROOF OF (2.2): We abbreviate Y := Y(p) * SW and A := A(p) with

W as in (2.3). Let [ be an open family of subgroups of G , i.e. $" P

contains full conjugaey classes and if H < K , H ~ ~ , then K ~T �9

Assume A(~) = {a ~ A I Iso(a) ~ ~ } and f(~) : A(~) --, Y to be

already constructed. Let H be maximal in ~ \~ . By assumption ii) H

we know that Y is a Z _~-homology sphere of dimension, say k . By (~J I

attaching cells of type WH • D with 1 ~ (k-dim WH) = m to A(~) H

we get an (m-l)-connected WH-map h : B --P yH extending f(~)H . For

p-toral K/H < WH , K + H we have hK/H = f(~)K and by assumption

H,(hK)(p) is an isomorphism, since K is sub-p-total in Gp by (2.1).

Therefore the homology H,(Ch)(p ) of the mapping cone C h , being con-

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centrated in * ~ [m+l,k+l] is projective over H...(WH)( ~ by Olivers �9 ~ " H p ) theorem (1.4). Lemma (2.4) implies that the inclusion Y --�9 C h is

Z(p)-homologically trivial. We want to show that the short exact se-

quence

._~8�9 H,(B)(p) h �9 ~,(yH)(p) 0 --�9 H,+l(Ch)(p) --�9 0

splits as a sequence of H(WH)(p)-modules. We first look at the case

* = k . The map h splits Z(p)-linearly. Since Hk+l(Ch)(p ) is pro-

jective as Ho(WH)(p)-module and hence relatively injective, we can

choose this splitting to be Ho(WH)(p)-equivariant. If * + m , we have

~,(yH)(p) = 0 . Hence the splitting is H,(WH)(p)-linear.

As k ~ 3+g by (2.3.ii)), we can use theorem (1.3) to get a rela-

tively free WH-space C m A(~) H of dimension S k and a WH-map C _�9

which is a Z(p)-homology equivalence. The canonical pushout

G iNH A(~)H ' G iNH C

A(~) �9 A(~')

finally defines A(~') for the family ~' = (~U (H)) . Lemma (2.2.iii)

shows that only finitely many such constructions have to be done. �9

3. PROOF OF THEOREM (i.i).

Let X be a (T,n)-resolution of A with H,(X) finitely generated.

First step: With help of (2.2) we can assume that Y(p) only has sub-

p-toral isotropy groups. We want to construct a G -equivariant map P I/

Y(p) --�9 X * S(p) = X(p) . If ~(p) is a Z(p~-homology,, sphere of di-

mension k(H) = (n(H) + dim S(p)H+I) by assumption, we do not assume

xH(p) to be (k(H)-l)-connected. To apply obstruction theory we lo-

calize a 2-sphere with trivial G -action at the prime p and construct P

a map f : Y(p) * S 2 --�9 X(p) * S~p) . Note that

Iso(Y(p) * S 2) = Iso(Y(p)) U {Gp} .

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BAUER-SCHNEIDER

As in the proof of (2.2) the map f will be constructed inductively

over families ~ .

We start with the inclusion S 2 --~ S~p) . If f(~) is constructed for

a family ~" and H ~ Iso(Y(p)) \ ~ is maximal, we can extend f(~r) H

over the (k(H)+3)-dimensional space (Y(p) * $2) H , since xH(p) * S~p)

is (k(H)+2)-connected. Now we have two cases: If H is a p-Sylowsub-

group, we perhaps have to choose another extension of S 2 --~ S~p) , in

order to get f(~H a Z(p)-homology isomorphism. This uses equivariant

obstruction theory. We refer to [2], Chapter II, especially to the pro-

positions before the proof of the equivariant Hopf theorem. (At some

places there the coefficient ring Z has to be replaced by Z(p) . The

homomorphism t is then actually an isomorphism). This obstruction

theory uses that IWHI is prime to p and the assumption iii) of the

theorem. If H + I is not maximal, the resulting map is a Z(p)-homology-

isomorphism by Smith theory. �9

Second step: The H,(Gp)(p)-homomorphism f, splits: By Olivers theo-

rem the cone of f is projective over the Pontrjaginring. Thus we need

only show f, to be injective. We restrict the Gp-action to a K-action

for a subgroup K of order p . The localized map fK(p) is a homology

and (without loss of generality) homotopy equivalence, since we are free

to stabilize. We can extend a homotopy inverse to a K-equivariant map

the (n(H)+3)-skeleton of X(p) * S~p), . Composition with f over gives

a k-map Y(p) * S 2 --~ Y(p) * S~p) . By Smith theory this is a Z(p)-ho-

mology equivalence. Hence f, is injective. �9

Before we take the third step, we need a technical Lemma.

(3.1) LEMMA: Let G operate diagonally on the join Z * S of a G-

space Z and a representation sphere S . There is an automorphism

of R such that suspension o : H.(Z) --~ H~,(Z * S) is a C-isomor-

phism, i.e. o(~(a) �9 z) = a �9 o(z) .

PROOF: Look at the selfmap of G x S , sending (g,s) to (g,g.s) .

The equality ~(a) x [S] = f,(a • [S]) defines ~ , where IS] de-

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BAUER-SCHNEIDER

notes the fundamental class of S . Now ~ is multiplicative: let

denote multiplication G x G --, G and T interchange the factors of

G x G . Then the identity

f(p • ids) = (p x ids)(T x ids)(id G x f)(T x ids)(id G x f)

gives ~(a'b) = t~(a) �9 @(b) .

Now let 0 : G x Z --, Z and 6 : G x Z x S --~ Z x S denote the

actions, T the interchanging Z x S -~ S x Z . Let 3 :H.+I(X*S)-~Ii.(XxS)~ ,~

denote the boundary map in the Mayer-Vietoris seuqence. The identity

= (g x ids)(id G x T)(f x idz)(idG x T) then gives:

a(a �9 o(z)) = (~(a) �9 z) • [s]

Taking into account the formula 8(a.o(z)) = (-l)lal(axSo(z)) , we see

that ~(a) = (-l)Ikl~(a) gives the desired automorphism of R.

Third step: Recall the abbreviations: R = H,(G) ; R ~ = H,(G o) ;

F = G/G . We want to analyse the R-module H,.,X = M . Let Q c M be the O ""

submodule R | Qn-g = R | Hn_g(X) . By the first two steps and (3.1)

we know that Q(p) is a projective R(p)-module, in particular (Qn_g)(p)

is a projective Z( )F -module for all p . Hence by Rim's theorem [5], P P

O Qn-g is projective as TF-module. Furthermore Q(p) ~ R(p) | Qn for

all p ~T x , hence Q -- R~ | Qn " This shows Q to be a projective

R-module.

Looking at the cokernel N = M/Q , we notice by the first two steps

N(p) ~ Z(p) . Since N is finitely generated by assumption i), we have

N ~ T , if we forget the R-action. As in the proof of (2.2) this gives

a direct sum decomposition M ~ Q | N . Now again we apply theorem (1.3)

to get the desired spherical (T,n)-resolution of A .

REMARK: The paper of Oliver would suggest to look only on p-toral sub-

groups, not on sub-p-toral ones. But we cannot generalize (2.1 i and

iii)) to this case.

The case of finite groups has been treated by tom Dieck in [2a].

The basic idea of the modification theorem (1.3) is being found already

in [7], section 2.

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BAUER-SCHNEIDER

REFERENCES

[i] BAUER, S.: Dimensionsfunktionen yon Homotopiedarstellungen kom-

pakter Liescher Gruppen, Dissertation, GSttingen 1986

[2] TOM DIECK, T.: Transformation groups, de Gruyter 1986

[2a] TOM DIECK, T.: Homotopiedarstellungen endlicher Gruppen: Dimen-

sionsfunktionen. Invent. Math. 67 (1982)

[3] DOTZEL, R.M.: An Artin relation (mod 2) for finite group actions

on spheres. Pacific J. Math. 114, 335-343 (1984)

[4] OLIVER, R.: Smooth compact Lie group actions on disks. Math. Z.

149 , 79-96 (1976)

[5] RIM, D.S.: Modules over finite groups, Ann. of Math. 69, 700-712

(1959)

[6] SCHNEIDER, A.: Dissertation (in preparation)

E7] SWAN, R.: Periodic resolutions for finite groups. Ann. Math. 72

(1960)

Stefan Bauer Sonderforschungsbereich 170 Geometrie und Analysis Bunsenstr. 3-5 D-3400 G6ttingen

Albert Schneider Mathematisches Institut Bunsenstr. 3-5 D-3400 GSttingen

(Received December 10, 1986; in rev i sed form October 5, 1987)

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