Lifting Problems and Transgression for Non-Abelian Gerbes

Details Lifting Problems and Transgression for Non-Abelian Gerbes Lifting Problems and Transgression for Non-Abelian Gerbes

2011-12-20

arxiv.org/abs/1112.4702v1

Mathematics

Algebraic Topology

33 pages

Scientific paper

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces - in case of a Lie 2-group with a single object - to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: Firstly, we obtain a new comparison theorem for different notions of string structures. Secondly, our transgression map establishes a direct relation between string structures and spin structure on the loop space.

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