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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