Lifting Problems and Transgression for NonAbelian Gerbes
Abstract
We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in nonabelian cohomology. We use our geometrical formulation in order to define a transgression map in nonabelian cohomology. This transgression map relates the degree one nonabelian cohomology of a smooth manifold (represented by nonabelian gerbes) with the degree zero nonabelian 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 2group with a single object  to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structure on the loop space.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 DOI:
 10.48550/arXiv.1112.4702
 arXiv:
 arXiv:1112.4702
 Bibcode:
 2011arXiv1112.4702N
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 55R65 (Primary) 18G50;
 22A22;
 53C08;
 55N05 (Secondary)
 EPrint:
 33 pages. v2 contains several small improvements