  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.
