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 S2. We also study the wedge sum of Eilenberg-MacLane spaces.