A Controlled Mather Thurston Theorem

The "c-principle" is a cousin of Gromov's h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that  for the MT-theorem, when the base dimensions is not equal four, only the mildest cobordisms, semi-s-cobordisms, are required. In a physical context, the extra topology (a perfect fundamental group) these cobordisms introduce could be hidden in the UV, suggesting that the action for gauge theories might be formulated in terms of group extensions and flat connections, rather than curvature. See arXiv:2006.00374  for some preliminary results.