Index and small bundle gerbes

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.