Equivariant $KK$-theory of $r$-discrete groupoids and inverse semigroups

TL;DR

This paper establishes an isomorphism between groupoid-equivariant and inverse semigroup-equivariant KK-theory, introduces descent homomorphisms, and discusses a Baum--Connes map for inverse semigroups, linking recent findings to KK-theory.

Contribution

It introduces a new isomorphism linking groupoid and inverse semigroup KK-theories and develops related descent homomorphisms and Baum--Connes map for inverse semigroups.

Findings

Isomorphism between ${ m G}$-equivariant and $S$-equivariant KK-theory

Definition of descent homomorphisms for inverse semigroups

Reflection of Khoshkam and Skandalis' results in KK-theory

Abstract

For an $r$-discrete Hausdorff groupoid ${\cal G}$ and an inverse semigroup $S$ of slices of ${\cal G}$ there is an isomorphism between ${\cal G}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define descent homomorphisms for $S$, and indicate a Baum--Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in $KK$-theory.