A characterization of equivariant absolute extensors and the equivariant Dugundji theorem

Abstract

Let G be a compact Lie group. We prove that a metrizable G-space X is a G-ANE (resp., a G-AE) iff X is an ANE (resp., an AE) and, for any closed subgroup H C G, the H-fixed point set X[H] is a strong neighborhood H-deformation retract (resp., a strong H-deformation retract) of X. If a G-space X has no G-fixed point, then X is a G-ANE provided that X is an H-ANE for any subgroup H C G that occurs as a stabilizer in X. As an application, we give a new proof of the equivariant Dugundji extension theorem in the metrizable case.