is in particular a foliation, the leaves being the fibers. *This will be the foliation of interest for us*. The action of $G$ on $W$ lifts to $M$, mapping leaves to leaves. Connes and Moscovici then construct a hypoelliptic signature operator on $M$ almost invariant under the $G$-action, in the sense that its Heisenberg leading symbol is $G$-invariant. This yields a regular spectral triple over the algebra ${C^{\infty}_c}(M)\rtimes G$, whose Chern-Connes character may be computed by means of the Connes-Moscovici residue formula ([@CM1995]). However, this does not directly provide a characteristic class formula, since the actual calculations give thousands of terms already for very low-dimensional manifolds $W$. To overcome this difficulty in higher dimensions, Connes and Moscovici developed cyclic cohomology for Hopf algebras in [@CM1998]. They defined such an algebra ${\mathcal{H}}$, which acts like a symmetry group allowing to reorganize the calculations, and built a characteristic map $\chi : {\mathrm{HP}}^{\bullet}_{\text{Hopf}}({\mathcal{H}}) \longmapsto {\mathrm{HP}}^{\bullet}({C^{\infty}_c}(M) \rtimes G)$. Then they prove

*(Connes-Moscovici, [@CM1998])* If the lifted action of $G$ on $M$ has no fixed points, then the Chern-Connes character of the hypoelliptic signature operator lies in the range of the characteristic map $\chi$.

In some sense, the group ${\mathrm{HP}}^{\bullet}_{\text{Hopf}}({\mathcal{H}})$ contains the geometric cocycles : 