8 GIORA DULA AND REINHARD SCHULTZ

Specifically, since W(Kj) acts freely on Fix(Kj,X) — SingFix(if7-,X) there is a

canonical isomorphism

HZiVj/G-V^/GiAiKj))*

BRH*N{KJ)

(FiK(Kj,X) - S i n g F i x ^ X ) ; ^ ^ ) )

(compare [tD, Section 2.9]), and one can combine this with the canonical iden-

tifications

Vj - Vj.! = G xN(Kj) (FXK(KJ,X) - SingFix(^,X))

BRH*N{Ki)iC(Y;A\N(Kj)) * BRH*G(G xN{Kj) Y;A)

to obtain an isomorphism

BRHZJVJ - V-_xM) * H: (VJ/G - Vj-xIG^Kj)).

Combining this with the preceding discussion, we obtain the desired isomorphism

from

BRH^VJ, VJ-I;A)

to # * (Fix*(i^-,X) - SingFix*(i^,X)M(i^)).B

The preceding results give a two step method for computing the homotopy

groups of equivariant function spaces by spectral sequences; the spectral sequence

of Theorem 1.4 is obtained from the filtration by orbit types, and the equivariant

Barratt-Federer spectral sequence is obtained from the filtration by skeleta. One

can combine these filtrations using refinements of ideas due to I. Fary [Fa] to

obtain a single spectral sequence for these homotopy groups. The details are

presented in [Sc2] with a sign correction in [M0, p. 116].

THEOREM

1.5. Let X be a finite simplicial complex with a simplicial action

of the finite group G, and let Y be a G-CW complex satisfying the Default Hy-

pothesis 1.0. Choose an indexing {(Ki)} for the conjugacy classes of isotropy

subgroups such that i j if a representative for (Kj) is contained in a represen-

tative for (Ki), let Ft be the G-subcomplex of points whose isotropy subgroups

represent (Kf), and let Xe = F\ U • • • U Fs. Then there is a spectral sequence

such that

EL c

0

f f i

{Xi.p/G,Xi.p.1/G;irlH.q.i (Fix(JQ_p,y)))

i

(the coefficients on the right may be twisted), with equality if p + q 2, and

such that E^q gives a series for 7rp+q(Fc(X, Y))M

As in Theorem 1.4 one can generalize to cases where X is an arbitrary finite

complex, but in these cases one should replace the relative cohomology groups

of (Xe/G,Xe-i/G with twisted coefficients by cohomology groups of XijG —

Xe-i/G with compact supports and twisted coefficients.

Application to smooth G-manifolds. Eventually we want to apply the pre-

ceding machinery to compact smooth G-manifolds; of course, this requires an