Mathematics Preprints 2008
08.04 : BROWN, R.
Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups
Summary:The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
Published in:Journal of Homotopy & Related Structures 3 (2008) 331-342.
Crossed complexes and higher homotopy groupoids as noncommutative tools for higher dimensional local-to-global problems
Abstract:We outline the main features of the definitions and applications of crossed complexes and cubical omega-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types in non simply connected situations.
(This is a revised version of a paper published in Fields Institute Communications 43 (2004) 101-130, which was an extended account of a lecture given at the meeting on `Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories', Fields Institute, September 23-28, 2002.)