Equivariant $\KK$-theory for generalised actions and Thom isomorphism in groupoid twisted $\K$-theory

TL;DR

This paper develops a new form of equivariant KK-theory for groupoid actions, introduces Stiefel-Whitney classes for vector bundles over groupoids, and establishes a Thom isomorphism in twisted groupoid K-theory.

Contribution

It extends equivariant KK-theory to general groupoid actions and proves a Thom isomorphism in twisted groupoid K-theory, incorporating Stiefel-Whitney classes.

Findings

Established equivariant KK-theory for groupoid actions.

Introduced Stiefel-Whitney classes for groupoid vector bundles.

Proved Thom isomorphism in twisted groupoid K-theory.

Abstract

We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.