Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

By Alain Connes, H. Moscovici. 2001

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fe3c3669d6869ccd5c1f2fc3cfb16679affd48b

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Citations in the corpus

Modular Hecke Algebras and their Hopf Symmetry (2003, methodology): “According to the two-step denition in [12], [ 14 ] of the isomorphism 1, one rst turns the Lie algebra cocycle gv 2 C2(a1; R) into a group 1-cocycle C1;0(gv) on G = Di + (R) with values in currents on J 1 +(R) , and then one”