Crossed Complexes and Higher Homotopy Groupoids as Noncommutative Tools for Higher Dimensional Local-to-Global Problems

Abstract

Publisher Summary This chapter presents a survey and explains the origins of results of crossed complexes obtained by R. Brown and P.J. Higgins and others over the years 1974–2008 and its applications and related areas. These results yield an account of some basic algebraic topology on the border between homology and homotopy. It differs from the standard account using crossed complexes, rather than chain complexes as a fundamental notion. In this way, not only classical results such as the Brouwer degree and the relative Hurewicz theorem but also noncommutative results on second relative homotopy groups as well as higher dimensional results involving the fundamental group through its actions and presentations are obtained comparatively quickly. One of the major results is a homotopy classification theorem that generalizes a classical theorem of Eilenberg–Mac Lane, though this does require results on geometric realizations of cubical sets. A replacement for the excision theorem in homology is obtained by using cubical methods to prove a Higher Homotopy van Kampen Theorem for the fundamental crossed complex functor on filtered spaces.