We define various classes of Sobolev bundles and connections and study their topological and analytical properties. We show that certain kinds of topologies (which depend on the classes) are well-defined for such bundles and they are stable with respect to the natural Sobolev topologies. We also extend the classical Chern-Weil theory for such classes of bundles and connections. Applications related to variational problems for the Yang-Mills functional are also given.
"Topological and analytical properties of Sobolev bundles. II. Higher dimensional cases." Rev. Mat. Iberoamericana 26 (3) 729 - 798, September, 2010.