Index and small bundle gerbes

TL;DR

This paper constructs small bundle gerbes over certain manifolds, develops an index theory for them, and applies it to identify obstructions to positive scalar curvature metrics.

Contribution

It generalizes the construction of decomposable bundle gerbes and establishes an Atiyah-Singer type index theorem for these structures.

Findings

Constructed small bundle gerbes over compact oriented aspherical 3-manifolds and higher dimensions.

Developed an index map related to fibrewise pseudodifferential and semiclassical smoothing operators.

Proved an Atiyah-Singer type theorem realizing push-forward into twisted K-theory.

Abstract

By a small bundle gerbe we mean a bundle gerbe in the sense of Murray defined on a smooth, finite-dimensional, fibre bundle over a manifold. We construct such gerbes over compact oriented aspherical 3-manifolds, as well as in higher dimensions, generalizing the construction of decomposable bundle gerbes in earlier work with Singer. For these small bundle gerbes there is a direct index map given in terms of either fibrewise pseudodifferential operators, or more conveniently fibrewise semiclassical smoothing operators, twisted by the simplicial line bundle. We prove the Atiyah-Singer type theorem that this realizes the push-forward into twisted K-theory. We also give an application via the index of projective families of Spin_c Dirac operators, to show the existence of obstructions to metrics with large positive scalar curvature.