A higher index theorem on manifolds with boundary

Abstract

The Atiyah-Singer index theorem expresses the Fredholm index of elliptic differential operators on compact manifolds in terms of geometric and topological data. The Atiyah-Patodi-Singer index theorem is a generalisation of this to compact manifolds with boundary. The contribution from the boundary in this theorem is the eta-invariant: a measure of the extent to which the spectrum a differential operator on the boundary is not symmetric with respect to reflection in 0. With Wang and Wang we obtained a generalisation of this result to a higher, K-theoretic index in the setting where a locally compact group acts on a possibly noncompact manifold with boundary, with compact quotient. This involves a higher, or equivariant version of the eta-invariant. Related results were obtained by Xie-Yu, Piazza-Posthuma-Song-Tang and others. This talk will start with a general introduction to K-theoretic index theory, in particular the type involving Roe algebras.