Fiber bundles in diffeology
At the CRCG Workshop - Higher Structures in Topology and Geometry III, June 4-5 2009.
I will give a short introduction about diffeological spaces, and try to underline the significant aspects in comparison with classical differential geometry. Then, I'll introduce the notion of fiber bundles in diffeology and give a few examples and properties (exact homotopy sequence, coverings, monodromy theorem...). I will apply these construtions to build the integration bundle p : Y -> X for any diffeological space X equipped with a closed 2-form w. It is a bundle with structure group the torus of periods of w, T = R/P , as soon as the periods P are a (diffeologically) discrete subgroup of the reals. This result uses the theory of differential forms in diffeology as well as the theory of fiber bundle. Actually the integration bundle comes with a connexion 1-form, for which w is the curvature, I'll discuss this point. And I'll be open to discuss any question concerning diffeology, homotopy theory, De Rham calculus, fiber bundles etc.