DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 9

interpretation of the latter as G — CW complexes. There are two ways of do-

ing this. Results of A. Wasserman [Wa] yield a version of Morse Theory for

G-invariant smooth functions on smooth G-manifolds, and from this it is ele-

mentary to show that smooth G-manifolds have the G-homotopy type of finite-

dimensional G — CW complexes; in fact, for compact smooth G-manifolds one

can choose the G — CW complexes to be finite (also see [Ko] for a self-contained

account of these results). On the other hand, for many purposes it is more useful

to use the smooth triangulation theorem due to S. Illman [IL2].

Extensions to equivariant diagram categories

If D is a small category and S is some category of topological spaces, then a D-

diagram with values in S is merely a covariant functor D — S and a morphism

of ^-diagrams is a natural transformation of functors. Results of W. Dwyer

and D. M. Kan [DK] show that one can extend much of classical homotopy

theory and obstruction theory to categories of P-diagrams with values in the

category of CW complexes. In this paper we shall need an equivariant version

of this theory. This can be constructed directly by combining the techniques

of equivariant obstruction theory with those of [DK], but everything we need

has been done by E. Dror Farjoun [DF] in greater generality. In the setting of

[DF] a G-space is viewed as a diagram on the category Cat(G) with one object

and morphisms given by the group structure of G; the usefulness of studying

equivariant topology by means of such diagrams had been shown in earlier work

of A. Elmendorf

[ELM].

We are interested in finite diagrams of subcomplexes of

a G-CW complex that correspond to partially ordered sets. In the spirit of [DF]

these may be viewed as diagrams on a product category Cat(G) x Cat(P), where

Cat(P) is the usual category associated to a partially ordered set P. Since we

are interested in subcomplexes of some larger complex, the partially ordered set

will usually have a unique maximal element.

Given a small category D, the constructions in [DF, Sections 4-5] yield gen-

eralizations of singular homology and cohomology to B-diagrams of spaces and

suitably defined coefficients. The latter are given by a functor from an orbit

category Orb(D) to modules over some appropriate ring. It follows from the

definitions in [DF, Section 1] that

Orb(Di x D2) = Orb(Oi) x Orb(D2)

and Orb(Cat(G)) is equivalent to the usual orbit category Orb ^ introduced

before (compare [DF, statements 1.5 and 1.7, pp. 97-98]). As noted in [DF,

statement 1.8], the orbit category Orb(P) can be very unpleasant even if D is

the well known category

2:={b-^c}

(i.e., two objects with one morphism from the first to the second) and this means

that the cohomology coefficient functor

M : Orb£f x Orb(Cat(P)op) —• Modules