Nonabelian algebraic topology

Abstract

This is a survey of central results in nonabelian algebraic topology. We present how the homotopy category of homotopy \(n\)-types and a certain localization of the category of crossed \(n\)-cubes of groups are equivalent. The functor inducing this equivalence satisfy a generalized Seifert-van Kampen theorem, in that it preserves connectivity and colimits of certain diagrams of generalized fibrations. We show descriptions of certain colimits of crossed \(n\)-cubes of groups and show how they have been used to generalize the Blakers-Massey theorem, the Hurewicz theorem and Hopf’s formula for the homology of groups, as well as a combinatorial formula for the homotopy groups of the sphere \(\mathbb {S}^2\). We also study the wedge sum of Eilenberg-MacLane spaces.