Equivariant diagrams of spaces

Abstract

We generalize two classical homotopy theory results, the Blakers–Massey theorem and Quillen’s Theorem B, to $G$–equivariant cubical diagrams of spaces, for a discrete group $G$. We show that the equivariant Freudenthal suspension theorem for permutation representations is a direct consequence of the equivariant Blakers–Massey theorem. We also apply this theorem to generalize to $G$–manifolds a result about cubes of configuration spaces from embedding calculus. Our proof of the equivariant Theorem B involves a generalization of the classical Theorem B to higher-dimensional cubes, as well as a categorical model for finite homotopy limits of classifying spaces of categories.