One of the central themes in algebraic topology is the use of homology and
cohomology groups to describe homotopy classes of continuous maps from one
space to another. The development of obstruction theory in the late nineteen
also see [Wh3], pp. 3-5) was a particularly important step in
this direction. When combined with later refinements, this provides a general
approach to analyzing homotopy classes of maps by cohomology invariants. Fur-
thermore, in many cases obstruction-theoretic techniques yield quantitative an-
swers to questions about homotopy classification that arise naturally in topology
(e.g., see [MT], pp. 140-142).
This paper develops an extension of classical obstruction theory to a cate-
gory of spaces with actions of a finite group G and maps that are isovariant
in the sense of R. S. Palais [P, 1.1.15, p. 6]; specifically, the maps in question
commute with the actions of G on the domain and codomain, and they also
preserve the isotropy subgroups of elements that fix the various points in the
domain and codomain. Although there are various uses for isovariant maps in
transformation groups, computational questions in isovariant homotopy theory
have generally been left unanswered. In this paper we shall show that isovariant
homotopy theory is equivalent to a special case of equivariant homotopy theory
for diagrams, at least for a restricted but interesting class of G-spaces. It follows
that one can use the work of W. Dwyer and D. Kan [DK] and E. Dror Farjoun
[DF; also see DF/Z] on categories of diagrams to study computational ques-
tions in isovariant homotopy theory by refinements of the standard techniques.
In particular, one has an obstruction theory whose cohomological invariants lie in
diagram cohomology groups. This theory is related to existing obstruction the-
ories for equivariant maps [Brel,M0,Sc2] in a natural manner. Furthermore,
the isovariant obstruction theory (in the cases covered) has the usual finite com-
putability properties as ordinary and equivariant obstruction theories. We shall
illustrate this by working out some examples that arise naturally in connection
with classification questions for differentiable group actions on spheres [Sc91].
Background information
Isovariant maps were used by Palais in [Pa] to generalize the standard clas-
sification theorems for free G-spaces when G is a compact Lie group; of course,
a free action is classified by the equivariant homotopy class of a map from the
space into a universal free G-space, and by Palais' results an arbitrary action with
finitely many orbit types is classified by the isovariant homotopy class of a map
from the space into a more elaborate universal G-space. The subsequent work of
A. Haefliger on embedding theorems for manifolds [Hael—2] and Browder and
Received by the Editors October 22, 1991, and in revised form March 22, 1993.
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