Tautological classes and higher signatures

Abstract: 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.