Obstruction theoretic methods are introduced into isovariant homotopy the-
ory for a class of spaces with group actions; the latter includes all smooth actions
of cyclic groups of prime power order. The central technical result is an equiv-
alence between isovariant homotopy and specific equivariant homotopy theories
for diagrams under suitable conditions. This leads to isovariant Whitehead the-
orems, an obstruction-theoretic approach to isovariant homotopy theory with
obstructions in cohomology groups of ordinary and equivariant diagrams, and
qualitative computations for rational homotopy groups of certain spaces of iso-
variant self maps of linear spheres. The computations show that these homotopy
groups are often far more complicated than the rational homotopy groups for the
corresponding spaces of equivariant self maps. Subsequent work will use these
computations to construct new families of smooth actions on spheres that are
topologically linear but differentiably nonlinear.
Key words and phrases. Barratt-Federer spectral sequence, Bredon cohomology, diagram
category, diagram cohomology, equivariant normal bundle, equivariant map, equivariant ho-
motopy, function space, group action, homotopy group, isovariant map, isovariant homotopy,
obstruction theory, quasistratification, treelike isotropy structure, Thorn-Mather stratification.
1991 Mathematics Subject Classification. Primary: 55P91, 55S37, 55S91, 55T99, 57R91,
57S15, 57S17. Secondary: 55N25, 55N35, 55P60, 55Q20, 55Q91, 57N80, 57Q91.