Specifically, since W(Kj) acts freely on Fix(Kj,X) SingFix(if7-,X) there is a
canonical isomorphism
(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-
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
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].
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
f f i
{Xi.p/G,Xi.p.1/G;irlH.q.i (Fix(JQ_p,y)))
(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
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