Equivariant diagrams of spaces

TL;DR

This paper extends fundamental homotopy theory results to G-equivariant diagrams of spaces, providing new tools for understanding equivariant homotopy limits and applications to G-manifolds.

Contribution

It generalizes the Blakers-Massey and Quillen's Theorem B to G-equivariant cubical diagrams, linking these to the equivariant Freudenthal suspension Theorem and configuration spaces.

Findings

Proves equivariant Blakers-Massey Theorem for G-diagrams

Derives equivariant Freudenthal suspension Theorem from Blakers-Massey

Generalizes Theorem B to higher-dimensional cubes

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.