A vanishing theorem for characteristic classes of odd-dimensional manifold bundles

We show how the Atiyah-Singer family index theorem for both usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle E -> X with odd-dimensional fibres, the family index ind (B) is an element of K-1 (X) of the odd signature operator is trivial. The Atiyah-Singer theorem allows us to draw a topological conclusion: the generalized Madsen-Tillmann-Weiss map alpha : B Diff(+) (M2m-1) -> Omega(infinity) MTSO (2m - 1) kills the Hirzebruch L-class in rational cohomology. If m = 2, this means that alpha induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen-Weiss theorem is wrong. For 3-manifolds M, we also prove the triviality of alpha in mod p cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.