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

Abstract

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.

arXiv comment10 pages, xypic, hyperref 25/06/08 version 2: 12 pages, accepted for JHRS, various minor revisions

References in corpus (3)

• The Fundamental Crossed Module of the Complement of a Knotted Surface
• Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras
• Normalisation for the fundamental crossed complex of a simplicial set

Cited by in corpus (4)

• Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
• Crossed modules and the homotopy 2-type of a free loop space
• Representing Bredon cohomology with local coefficients
• Covering morphisms of crossed complexes and of cubical omega-groupoids with connection are closed under tensor product