Quinn's stratified surgery theory [BQ] provided further uses of isovariant maps
in geometric topology.
The original motivation for this work came from differentiable transforma-
tion groups and the second named author's search for a unified setting for the
Browder-Quinn stratified surgery theory [BQ] and classification theorems of
Browder-Petrie [BP] and Rothenberg-Sondow [RS] for certain semifree differen-
tiable group actions on homotopy spheres (see [Sc84], Section 5, for a summary).
Some preliminary results along these lines were described in an unpublished an-
nouncement [Sc76]; these include (i) the construction of a structure set theory
for smooth G-manifolds within a fixed almost isovariant homotopy type using
techniques of [BQ] and almost isovariant homotopy-theoretic invariants, (ii)
modifications of the latter to study arbitrary smooth abelian p-group actions on
homotopy spheres (see [Sc87] for a more recent and accessible summary). A
detailed account of these topics is planned for future papers (e.g., see [Sc91]).
It seems likely that isovariant homotopy has other uses in geometric topology
and should provide new information on equivariant surgery in situations where
the latter's standard general position assumption (i.e., the Gap Hypothesis [DS,
Section 1.4]) does not hold.
Overview of the paper
This paper is divided into seven sections. The first discusses the approaches
to equivariant obstruction theory in work of Bredon [Brel], M0ller [M0], and
the second named author of this paper [Scl—2] together with their extensions
to diagram categories. In the Sections 2-4 we construct the formal setting for
our work. Our main interest lies with the category of compact differentiable
G-manifolds (where G is a finite group) and isovariant maps. The objects of
this category have very well behaved decompositions of various types, includ-
ing equivariant triangulations [IL2] and equivariant Thorn-Mather stratifications
(see [DS, Section 2.4]). The features that are important to our work are car-
ried by a structure we shall call a quasistratification. The basic properties of
these structures are presented in Section 2, and in Section 3 we show that an
isovariant map of compact differentiable G-manifolds can always be deformed to
be compatible with canonical quasistratifications on the domain and codomain.
Furthermore, if the isotropy subgroup structure is treelike in the sense of Section
3, then we can deform the map to have even stronger compatibility properties.
The class of smooth group actions with treelike isotropy structure is large enough
to contain many interesting types of G-manifolds, and it is convenient to work
with because there are no problems with intersections of fixed point sets of dif-
ferent subgroups (either two such subsets are disjoint or one is contained in
the other). In Section 4 we introduce the notion of an almost isovariant map.
These maps are compatible with quasistratifications and may fail to be isovari-
ant only on tightly controlled subsets of the domain; for many interesting and
useful classes of smooth G-manifolds one can deform almost isovariant maps to
isovariant maps by almost isovariant homotopies (one can also do this relative
to a well behaved subset on which the original map is already isovariant, at least
in a weak sense). Since almost isovariant maps are defined to be morphisms
of certain diagrams, it follows immediately that one has reasonable obstruction
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