A vanishing theorem for characteristic classes of odd-dimensional manifold bundles (Q2865900)

The groundbreaking work of Madsen, Tillmann, Weiss, et al. [See, as one example among many, \textit{S. Galatius} et al. [Acta Math. 202, No. 2, 195--239 (2009; Zbl 1221.57039)] begins with the following construction. Let \(M\) be a closed smooth oriented manifold and \(E \to B\) a smooth oriented \(M\)-bundle over a paracompact base space \(B\). Then there is a vertical tangent bundle \(T_vE\) over \(E\); taking the negative of this bundle yields a map of Thom spectra \(\kappa_E:\mathbf{Th}(-T_vE) \to MTSO(n)\) where \(MTSO(n)\) is the Thom spectrum of the negative of the tautological oriented \(n\)-place bundle. Since \(B\) is paracompact, there is an embedding over \(B\) from \(E\) into \(B \times \mathbb{R}^\infty\) identifying \(-T_vE\) with a tubular neighborhood. The collapse map \(\Sigma^\infty B_+ \to \mathbf{Th}(-T_vE)\) is the Pontrjagin-Thom map. Composing this with \(\kappa_E\) and taking the adjoint gives a map NEWLINE\[NEWLINE \alpha_E: B \to \Omega^\infty MTSO(n). NEWLINE\]NEWLINE Much about the homotopy type of the target is well understood.NEWLINENEWLINEThe main theorems in this area vary \(B\) and \(M\) and analyze what happens in the limit. For example, \(B\) could be the universal example \(B\mathrm{Diff}^+(M)\), where the \(+\) denotes diffeomorphisms fixing a marked point. Using that marked point, one can then stabilize under connected sum with some fixed \(n\)-manifold. In the celebrated paper of \textit{I. Madsen} and \textit{M. Weiss} [Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)] the authors work at \(n=2\) and stabilize using the torus. The end result is a homology isomorphism between the classifying space of the stable mapping class group and \(\Omega^\infty MTSO(2)\).NEWLINENEWLINEThe case \(n=2\) is very special, partly due to homology stability results such as \textit{J. L. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)] and because the diffeomorphism groups of oriented \(2\)-manifolds of high genus are homotopically discrete. What happens in higher dimensions is more mysterious, although there have been recent successes in even dimensions. See, for example, \textit{S. Galatius} and \textit{O. Randal-Williams} [Acta Math. 212, No. 2, 257--377 (2014; Zbl 1377.55012)]. The aim of this paper is to analyze homological properties of \(\alpha_E\) in odd dimensions.NEWLINENEWLINEThe main theorem uses index theory to construct non-trivial classes in \(K\)-theory which must vanish under \(\alpha_E^\ast\) for all \(E\). Applying the Chern character yields maps in rational cohomology which also must vanish under \(\alpha_E^\ast\). After some analysis, the author identified this class in rational cohomology as a variant of the Hirzebruch \(\mathcal{L}\)-class. As a corollary the map \(\alpha_E^\ast\) in rational cohomology is identically zero in dimension \(3\).NEWLINENEWLINEThe author points to another paper \textit{J. Ebert} [Algebr. Geom. Topol. 11, No. 1, 69--105 (2011; Zbl 1210.55012)] for two additional assertions. First, that there is no analog of this result in even dimensions. Second, in odd dimensions this is ``the only vanishing theorem of this type.''