Isovariant extensors and the characterization of equivariant homotopy equivalences

Abstract

We extend the well-known theorem of James–Segal to the case of an arbitrary family F of conjugacy classes of closed subgroups of a compact Lie group G: a G-map f : X ! Y of metric EquivF-ANE-spaces is a G-homotopy equivalence if and only if it is a weak G-F-homotopy equivalence. The proof is based on the theory of isovariant extensors, which is developed in this paper and enables us to endow F-classifying G-spaces with an additional structure.