The codimension 2 index obstruction to positive scalar curvature

Abstract: We address the following general question: Given a (compact without boundary) manifold M, does M admit a metric of positive scalar curvature. Very classically, the Gauss-Bonnet theorem implies that among the (connected orientable compact) surfaces only the 2-sphere has this property.

In higher dimensions, the most powerful information uses the Dirac operator and its index, and an old observation of Schrödinger ("Über das Diracsche Elektron im Schwerefeld") coupling scalar curvature to the latter.

We will quickly introduce classical and more modern ("higher") index theory approaches to this problem, and then discuss a special implementation:

How and why certain submanifolds of codimension 2 act as a vaccine (or poison, depending on your point of view) and prevent the occurrence of positive scalar curvature metrics. Realistically, there won't be too much time to talk about that.

algebraic topologygeometric topologyoperator algebras

Audience: researchers in the topic

Online GAPT Seminar

Series comments: Description: Seminar of the GAPT group at Cardiff University