  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 $E \to X$ with
odd-dimensional fibres, the family index $\ind (B) \in K^1 (X)$ of the odd
signature operator is trivial. The Atiyah-Singer theorem allows us to draw a
topological conclusion: the generalized Madsen-Tillmann-Weiss map $\alpha: B
\Diff^+ (M^{2m-1}) \to \loopinf \MTSO(2m-1)$ kills the Hirzebruch $\cL$-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 $\alpha: B \Diff^+ (M) \to \MTSO (3)$ in mod $p$ cohomology in
many cases. We show an appropriate version of these results for manifold
bundles with boundary.
