Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

Abstract

We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this isomorphism can be explicitly implemented at the cochain level, by a cochain map constructed out of a fixed torsion-free linear connection. This allows the direct treatment of the index formula for the hypoelliptic signature operator - representing the diffeomorphism invariant transverse fundamental $K$-homology class of an oriented manifold - in the general case, when this operator is constructed by means of an arbitrary coupling connection.