Proper actions of locally compact groups on equivariant absolute extensors

Abstract

Let G be a locally compact Hausdorff group. We study equivariant absolute (neighborhood) extensors (G-AE's and G-ANE's) in the category G-M of all proper G-spaces that are metrizable by a G-invariant metric. We first solve the linearization problem for proper group actions by proving that each X is an element of G-M admits an equivariant embedding in a Banach G-space L such that L\{0} is a proper G-space and L\{0} is an element of G-AE. This implies that in G-M the notions of G-A(N)E and G-A(N)R coincide. Our embedding result is applied to prove that if a G-space X is a G-ANE (resp., a G-AE) such that all the orbits in X are metrizable, then the orbit space X/G is an ANE (resp., an AE if, in addition, G is almost connected). Furthermore, we prove that if X is an element of G-M then for any closed embedding X/G hooked right arrow B in a metrizable space B, there exists a closed G-embedding X hooked right arrow Z (a lifting) in a G-space Z is an element of G-M such that Z/G is a neighborhood of X/G (resp., Z/G = B whenever G is almost connected). If a proper G-space X has metrizable orbits and a metrizable orbit space then it is metrizable (by a G-invariant metric).