Hopf algebras, cyclic cohomology and the transverse index theorem

In this paper we solve a longstanding internal problem of noncommutative ge ometry, namely the computation of the index of transversally elliptic opera tors on foliations, We show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very s pecific Hopf algebra H-n, associated to each integer codimension, This Hopf algebra reduces transverse geometry, to a universal geometry of affine nat ure. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper, in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the Hochschild c ohomology of a Hopf algebra carries a highly nontrivial cyclic structure.