A vanishing theorem for characteristic classes of odd-dimensional manifold bundles
By Johannes Ebert. 2009
55 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/057c16f8ea6d5115e8a751b64fa3193bb2948a33
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In Algebra and Number Theory
In Mathematical Physics
- O. Randal-Williams
- Johannes Ebert
- B. Botvinnik
- Søren Galatius
- Nathalie Wahl
- Andrew Putman
- Steven V. Sam
- Benson Farb
Citations in the corpus
- Infinite loop spaces and positive scalar curvature (2014, methodology): “he maps ind(E),(Ω∞λ−d) ◦αE: X−→Ω∞+dKO are weakly homotopic. The translation of Theorem 3.7.2 into Theorem 3.8.2 is exactly parallel to the translation of the complex family index theorem described in [17]. For manifold bundles with nonempty boundary, we need apsc metric h∈R+(M) to be able to talk ab”
- Stable moduli spaces of high-dimensional manifolds (2012, background): “(The case k = Q was proved by Ebert.)”