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).