Universität Augsburg
Institut für Mathematik

Given a smooth manifold one can ask whether there is a metric of positive scalar curvature on it. The study of this existence question has produced many obstructions. The natural follow-up question is uniqueness of positive scalar curvature metrics. One way to phrase it is „What is the homotopy type of the space of psc metrics?". A central tool used in its study is the index difference by Hitchin, which provides a map to $K$-theory. This map is known to be non-trivial on homotopy groups if the dimension of the manifold is at least 6.

In my talk I will explain the methods used to prove this detection result and present new enhancements, which allow for a simultaneous extension to dimension 5 and incorporation of the fundamental group (yielding an index difference mapping to the $K$-theory of group $C^*$-algebras). These enhancements use the original Madsen–Weiss theory and its extension by Perlmutter, which has been dubbed "parametrised Morse theory". This is in contrast to methods of Galatius–Randal-Williams employed in earlier results.