Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen and suspension theorems

Abstract

Let G be a compact Lie group and V be a G-representation. We define V-dimensional equivariant Eilenberg-MacLane spaces and show that their elementary properties imply a Seifert-van Kampen theorem and a suspension theorem for the Vth homotopy groups of G-spaces. Our equivariant suspension theorem is radically different from those that have appeared previously. Rather than asserting, under certain very restrictive hypotheses, that the suspension map is an isomorphism (or epimorphism), our theorem describes, under milder hypotheses, the precise extent to which this map fails to be injective and surjective.