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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163483The 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.10 pages, xypic, hyperref 25/06/08 version 2: 12 pages, accepted for JHRS, various minor revisionsAlgebraic TopologyCategory Theory13D02,18G50,20J05,55S37,55S45Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groupstext