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

TL;DR

This paper proves a vanishing theorem for characteristic classes of odd-dimensional manifold bundles, showing that certain index invariants are trivial and that related topological maps induce zero in rational cohomology, challenging existing conjectures.

Contribution

It establishes a vanishing result for the family index of the odd signature operator in odd-dimensional bundles, linking index theory with the topology of moduli spaces of manifolds.

Findings

Family index of odd signature operator is trivial for odd-dimensional bundles.

The Madsen-Tillmann-Weiss map kills the Hirzebruch L-class rationally.

The 3-dimensional analogue of the Madsen-Weiss theorem does not hold in general.

Abstract

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.