  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.
