Simulating curvature with flat connections

Abstract:  Often a bundle may be coborded to a new one possessing a flat connection.  This is the subject of a paper of Milnor and Wood and in a much broader context the Mather-Thurston thm.  There are many open geometric questions about how “much damage” the cobordism needs to do to the original bundle. I’ll explain the questions and give some partial answers.

As usual, the motivation is physics: Can we hope to replace Maxwell terms in familiar Lagrangians like EM or standard model, with terms instead charging action for the presence of extra (fluctuating) topology in the base and distortion of transition functions in the bundle.