Cyclic cohomology of etale groupoids. (English) Zbl 0812.19003

The authors indicate three necessary steps in a program to extend Bismut’s local index theorem for families to foliations (in the sense of Connes’ index theory). The first step which defines the curvature of the index in terms of cyclic homology is touched on in the present paper. The next two steps, extension of Quillen’s superconnection formalism to operators with nondiscrete spectrum and the local computations, are left to future work. Here they determine the Hochschild and cyclic homology and the cyclic cohomology of the algebra of functions on a separated smooth étale groupoid (which is Morita equivalent to the relevant foliation groupoid and is therefore supposed to have the same cyclic cohomology). Applications of the main results on cyclic (co)homology are obtained by specializing the groupoid, in particular, to the transformation groupoid obtained from a smooth action of a discrete group on a manifold. This contains as a special case the groupoid of an orbifold, and leads to an interpretation of the orbifold Euler characteristic in cyclic homology. Another application gives an expression of the Chern character of a vector bundle over a smooth manifold in terms of a defining cocycle of transition maps.


19D55 \(K\)-theory and homology; cyclic homology and cohomology
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