Quotient projections as equivariant fibrations

Abstract

We will consider the canonical projection q: G-->G/H as an H-map with the action by conjugations and prove that it is an H-fibration provided G is an almost connected metrizable group and H is a compact subgroup of G. As a consequence, any G-map E-->G/H has the equivariant homotopy lifting property with respect to any G-space.