1. Equivariant homotopy in diagram categories

Our development of isovariant obstruction theory will be formally analogous

to the standard treatment of equivariant obstruction theory. More precisely, we

shall show that isovariant obstruction theory for maps of compact differentiable

G-manifolds is equivalent to equivariant obstruction theory for a suitable cat-

egory of equivariant diagrams. Therefore we shall begin by summarizing the

basic concepts from equivariant obstruction theory and indicating how their ex-

tensions to diagram categories can be extracted from work of W. Dwyer and D.

M. Kan [DK] and E. Dror Farjoun [DF]. Some relationships between the results

of Bredon [Brel], J. M0ller [M0], and the second author of this paper [Sc2] will

also be discussed, but we shall not attempt to cover all of equivariant obstruction

theory. Further references are given in [tD] and [Sc85, Section 1]. Terminology

not defined explicitly in this article can be found in the books of Bredon [Bre2]

and torn Dieck [tD].

In order to avoid bookkeeping difficulties we shall frequently make the follow-

ing assumption throughout this section:

DEFAULT HYPOTHESIS

1.0. The (finite) group G acts on the (Hausdorff)

space Y such that, for each subgroup H C G, the fixed point set of H in Y is

either empty or arcwise connected and simple.

As usual (compare [Wh2]), an arcwise connected space Y is said to be simple

if 71* 1 (y) acts trivially on nn(Y) for all n 0. We shall be particularly interested

in spaces of the form Y = A x S(V), where A is an arcwise connected simple

space with trivial G- action (e.g., a sphere or a disk) and S(V) is the unit sphere

in some finite-dimensional orthogonal G-representation V such that no fixed set

Fix(H, V) is 1-dimensional; all these spaces satisfy Hypothesis 1.0.

Review of equivariant obstruction theory

Ordinary obstruction theory for maps from a CW complex X to a second space

Y is formulated by using the singular cohomology groups of X with coefficients in

the homotopy groups of Y (see [Wh2] for details). By construction, equivariant

obstruction theory is formally analogous to the usual theory as developed in

[Wh2], the main difference being the need to use more elaborate equivariant

cohomology groups. In current terminology, the theory of [Brel] uses the Bredon

cohomology groups

BRHQ(X)

of a G-CW complex X. These are most simply

viewed from the cellular approach of [Brel], but subsequent work of Th. Brocker

[Bro] and S. Illman [ILI] shows that these can be defined as equivariant singular

cohomology groups with suitable equivariant coefficients. Usually the latter are

given by a contravariant functor

A : OrbG -* AbelGps

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