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

Ronald Brown

The classifying space of a crossed complex generalises theconstruction of Eilenberg-Mac Lane spaces. We show how the theory offibrations of crossed complexes allows the analysis of homotopyclasses of maps from a free crossed complex to such a classifyingspace. This gives results on the homotopy classification of mapsfrom a $CW$-complex to the classifying space of a crossed module andalso, more generally, of a crossed complex whose homotopy groupsvanish in dimensions between $1$ and $n$. The results are analogousto those for the obstruction to an abstract kernel in groupextension theory.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 331-342