DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 7
(2) If X is a finite-dimensional simplicial complex and G acts simplicially
with the regularity properties of [Bre2, Ch. Ill], then one can interpret the
cohomology group with compact supports as an ordinary relative cohomology
group. Specifically, let Nj be a suitably chosen simplicial regular neighborhood
of SingFix(Kj, X) in Fix(Kj, X) for as suitably fine barycentric subdivision,
let Nj be the interior of this neighborhood, and let dNj be its frontier. Under
these conditions the cohomology of Fix* (Kj, X) SingFix* (i^ , X) with compact
supports is just the relative cohomology of the pair
({Fix*(^-,X) - N^/WiK^dNj/WiKj)).
Similar considerations arise in [Sc2, proof of Thm. 4.3, pp. 82-84].
Proof. (Sketch) In principle this has been known to. workers in the area
for two decades, but neither the statement nor the proof appears to be in the
literature. Consider the filtration of X by the subsets
X3= \J Fix(i^,X),
Vj
= [jXr
If we take the long exact sequences in
BRHQ(—\
A) for the pairs (Vj, V{) we obtain
an exact couple whose Eoo term is a series for
BRHQ(X;A)
and whose E\ term
is given by
E\j =
BR
Hi+i{Vj,Vj„l;A).
Clearly it suffices to show that these groups are isomorphic to the ordinary
cohomology groups (with twisted coefficients and compact supports) that appear
in the statement of the theorem.
The first step is to notice that Vj-\ and Vj are subcomplexes of X. Since
a finite G-CW complex is a G-equivariant absolute neighborhood retract [Sc2,
Section 2], a straightforward extension of the standard tautness lemma [Sp, Thm.
10, p. 290] shows that the pair (Vj, Vj_i) is equivariantly taut with respect to
Bredon cohomology. Therefore if we define Bredon cohomology with compact
supports by
BRH^C(Y;
A) := cohm
B
RH*G(Y,
Y-L;A)
where L runs through all G-invariant closed subsets with compact closure, it
follows that
BRH^V^VJ^A)
^
BRH*G,C(VJ
- Vj^A).
Since G acts on ^ - Vj_i with the single orbit type G/Kj, the right side can
be rewritten in terms of the orbit space
(Vj - Vj-i) /G = Vj/G - Vj-i/G ^ Fix*(Kj, X) - SingFix*^-, X).
Previous Page Next Page