Abstract

What is the topology of the space of metrics of positive scalar curvature on a given manifold M? This question has received considerable attention in recent years. An old construction of Hitchin shows how one can use the action of the diffeomorphism group to construct intersting elements in this space, and use index theory to distinguish these. This allowed him to construct non-trivial components and classes in the fundamental group. A few years ago, in joint work with Diarmuid Crowley, we showed that one can obtain non-trivial homotopy classes of arbitrarily high degree. The main novelty lies in homotopy theory: we exploit the non-trivial product structure of K-theory and stable homotopy.

In the talk, we will describe this method and the work in progress which also covers the remaining half of degrees. This is based on the use of Toda brackets, a secondary product. Along the way, we get new information about the diffeomorphism group of spheres, in particular about its Gromoll filtration