Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups

Details Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups

2008-02-29

arxiv.org/abs/0802.4357v3

Mathematics

Algebraic Topology

10 pages, xypic, hyperref 25/06/08 version 2: 12 pages, accepted for JHRS, various minor revisions

Scientific paper

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.

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