Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

TL;DR

This paper establishes a cyclic cohomology analogue of Haefliger's theorem for diffeomorphism groupoids, revealing a Hopf algebraic structure and linking it to Gelfand-Fuchs cohomology, with applications to index formulas for hypoelliptic operators.

Contribution

It introduces a differentiable cyclic cohomology framework with a Hopf algebra structure and explicitly relates it to Gelfand-Fuchs cohomology via a cochain map, advancing transverse geometry analysis.

Findings

Proves a cyclic cohomological analogue of Haefliger's theorem.

Shows the algebra of transverse differential operators has a Hopf algebraic structure.

Establishes an explicit isomorphism with Gelfand-Fuchs cohomology and applies it to index formulas.

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.