Concrete cup and cap product formulas for ample groupoids are developed and applied to tiling-space cohomology computations and to an asymptotic-innerness invariant for C*-algebra automorphisms.
A Homology Theory for Etale Groupoids
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abstract
Etale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.
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Cup and cap products for cohomology and homology groups of ample groupoids
Concrete cup and cap product formulas for ample groupoids are developed and applied to tiling-space cohomology computations and to an asymptotic-innerness invariant for C*-algebra automorphisms.