Connes and Moscovici showed that it is isomorphic to the Gel’fand-Fuchs cohomology, which contains e.g characteristic classes of equivariant vector bundles over $M$. It then remains to actually compute the preimage of the Chern-Connes character. Explicit calculations are made in [@CM1998] in dimension 1, giving (twice) the transverse fundamental class of [@C83]. In dimension 2, the authors show that the coefficient of the first Pontryagin class does not vanish.\
Our Theorem \[intro thm equivariant index\] allows to short-cut the calculation with Hopf algebras and gives direct answer to the problem of computing the Chern-Connes character of the hypoelliptic signature operator in terms of equivariant characteristic classes, for manifolds $W$ of arbitrary dimension.

\[intro thm CM signature\] Let $G$ be a discrete group of orientation-preserving diffeomorphisms on a manifold $W$. Let $M$ be the bundle of Riemannian metrics over $W$. If the lifted action of $G$ has no fixed points on $M$, then the Chern-Connes character of the Connes-Moscovici hypoelliptic signature operator is $$\pi_*\circ\Phi(L'(M))\ \in HP^1(C^{\infty}_c(M)\rtimes G)\ ,$$ where $L'(M)$ is the modified $L$-genus, $\Phi: H^{\mathrm{ev}}(EG\times_G S^*_HM) \to HP^1(C^{\infty}_c(S^*_HM)\rtimes G)$ is Connes’ characteristic map, and $\pi_*: HP^1(C^{\infty}_c(S^*_HM)\rtimes G) \to HP^1(C^{\infty}_c(M)\rtimes G)$ is induced by the canonical projection $\pi : S^*_HM\to M$.

