An etale groupoid's invariant measures pair canonically with its zeroth Crainic-Moerdijk homology, yielding a groupoid Elliott invariant shown to match the C*-algebraic Elliott invariant for many integer actions and orbit-breaking constructions.
The rational HK-conjecture: transformation groupoids and a revised version
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abstract
We prove the rational HK-conjecture for a large class of transformation groupoids in the case when the relevant action has torsion-free stabilizers. A revised version of the rational HK-conjecture in the case of (possibly) torsion stabilizers is introduced and proved for a large class of transformation groupoids. In particular, this revised version holds for Scarparo's counterexamples to the original rational HK-conjecture. The key tools used are the Baum-Connes conjecture and a Chern character defined by Raven.
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A trace pairing and Elliott invariant for groupoid homology
An etale groupoid's invariant measures pair canonically with its zeroth Crainic-Moerdijk homology, yielding a groupoid Elliott invariant shown to match the C*-algebraic Elliott invariant for many integer actions and orbit-breaking constructions.