Two homotopy characterizations of G-ANR spaces for proper actions of Lie groups

Abstract

For a Lie group G we consider the class G-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G-invariant metric. It is proved that if each point x∈X of a proper G-space X admits a Gx-invariant neighbourhood U which is a Gx-ANE then X is a G-ANE, where Gx stands for the stabilizer of x. We give two equivariant homotopy characterizations of proper G-ANR spaces in the class G-M. One of them asserts that a G-space Y∈G-M is a G-ANR iff Y is locally G-contractible and every metrizable closed G-pair (X,A) with X∈G-M has the G-equivariant homotopy extension property with respect to Y (for short, property G-HEP). Another result establishes that a stronger homotopy property than G-HEP also characterizes G-ANR spaces in the class G-M.