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

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.

On the Thom Isomorphism Theorem

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).