For a bundle of oriented closed smooth $n$-manifolds $\pi: E \to X$, the tautological class $\kappa_{ \mathcal{ L}_k} (E) \in H^{ 4k-n}(X;\mathbb{ Q})$ is defined by fibre integration of the Hirzebruch class $\mathcal{ L}_k (T_v E)$ of the vertical tangent bundle. More generally, given a discrete group $G$, a class $u \in H^p(B G;\mathbb{ Q})$ and a map $f:E \to B G$, one has tautological classes $\kappa_{ \mathcal{ L}_k ,u}(E,f) \in H^{ 4k+p-n}(X;\mathbb{ Q})$ associated to the Novikov higher signatures. For odd $n$, it is well-known that $\kappa_{ \mathcal{ L}_k}(E)=0$ for all bundles with $n$-dimensional fibres. The aim of this note is to show that the question whether more generally $\kappa_{ \mathcal{ L}_k,u}(E,f)=0$ (for odd $n$) depends sensitively on the group $G$ and the class $u$. For example, given a nonzero cohomology class $u \in H^2 (B \pi_1 (\Sigma_g);\mathbb{ Q})$ of a surface group, we show that always $\kappa_{ \mathcal{ L}_k,u}(E,f)=0$ if $g \geq 2$, whereas sometimes $\kappa_{ \mathcal{ L}_k,u}(E,f)\neq 0$ if $g=1$. The vanishing theorem is obtained by a generalization of the index-theoretic proof that $\kappa_{ \mathcal{ L}_k}(E)=0$, while the nontriviality theorem follows with little effort from the work of Galatius and Randal-Williams on diffeomorphism groups of even-dimensional manifolds.
Categories: math.GT math.AT