DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 5
where Orb c is the so-called orbit category of homogeneous G spaces G/H and
equivariant maps between such spaces. In more concrete terms, this means one
has an abelian group A(H) for every subgroup H C G, and one also has an
action of
W(H) := NORMALIZERG(#)/#
on A(H) for all H. These are related by some standard compatibility rules; for
example, if H C K one has a natural map A{K) A(H). For equivariant
obstruction theory, the coefficient system of interest is an n-th homotopy group
functor G^n(y) such that
{G^TI(Y)}(H)
7rn(Fix(i7,Y)) and the action of
W{H) on the latter is induced by the natural action of W(H) on Fix(H, Y).
Our default hypothesis ensures that the objects under consideration are abelian
groups and the action of W(H) on 7rn does not depend upon the choice of a
basepoint.
Remark. The discussion above is not quite correct if some set Fix(iJ, Y) is
empty, but the modifications in such cases are fairly easy. Following the notation
of S. J. Willson [Wi] we shall say that a family of subgroups of a group G is a
list if it is closed under conjugation and passage to subgroups; such collections
are called open (isotropy) families of subgroups in [tD] {e.g., see p. 46). If Y
is a G-space let Hy be the set of all subgroups H C G such that Fix(H, Y) is
nonempty; then Hy is a list of subgroups in the sense of [Wi] and the homotopy
groups 7rn(Fix(iJ, Y)) form a system of equivariant coefficients over Hy in the
sense of [Wi]. The setting of [Wi] then provides an appropriate definition of
BRHQ(A;
G7rn(F)) if A is any space whose isotropy subgroups lie in Hy. Since
the isotropy subgroups for X lie in Hy if there is an equivariant map from X to
V, it follows that one can use the setting of [Wi] to define
BRHQ(X;
G^niY)) if
there is an equivariant map from X to Y.
As in [Brel], our default hypothesis allows us to avoid Bredon cohomology
with twisted coefficients. However, results of J. M0ller [M0] and I. Moerdijk and
J.-A. Svensson [MSv] provide treatments of equivariant twisted coefficients that
are adequate for developing equivariant obstruction theories without the simplic-
ity hypotheses in 1.0. It seems likely that one could combine the machinery of
[M0] and [MSv] with the techniques of this paper to circumvent our simplicity
hypothesis.
Ordinary obstruction theory relies heavily on the homotopy extension prop-
erty for the inclusion of a closed subcomplex A in a CW complex X. In particu-
lar, this implies that a continuous map A—+Y extends to X if and only if there
is an extension up to homotopy. Similarly, one has the G-homotopy extension
property for the inclusion of a closed G-subcomplex A in a G-CW complex X.
In fact, the proof is a simple variant of the argument in the nonequivariant set-
ting (compare [tD, Exercise 1.17.3, p. 103]; a more explicit reference is [Mat,
statement (J), p. 369]). This property and the machinery of Bredon cohomology
allow one to extend the formal setup of obstruction theory (as in [Wh2]) to the
category of G-CW complexes; the main results can be stated as follows:
THEOREM
1.1. (Extension Theorem) Let A be a G-subcomplex of the G-CW
complex X, and let f : A Y be a continuous equivariant map into a G-space
Previous Page Next Page