# Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey - Mathematics > K-Theory and Homology

Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey - Mathematics > K-Theory and Homology - Download this document for free, or read online. Document in PDF available to download.

Abstract: We survey work by the author and Ralf Meyer on equivariant KK-theory. Dualityplays a key role in our approach. We organize the survey around the objectiveof computing a certain homotopy-invariant of a space equipped with a properaction of a group or groupoid called the Lefschetz map. The Lefschetz mapassociates an equivariant K-homology class to an equivariant Kasparovself-morphism of a space X admitting a dual. We want to describe it explicitlyin the setting of bundles of smooth manifolds over the base space of a propergroupoid, in which groupoid elements act by diffeomorphisms between fibres. Toget the required description we describe a topological model of equivariantKK-theory by way of a theory of correspondences, building on ideas of PaulBaum, Alain Connes and Georges Skandalis that appeared in the 1980-s. Thismodel agrees with the analytic model for bundles of smooth manifolds under sometechnical conditions related to the existence of equivariant vector bundles.Subject to these conditions we obtain a computation of the Lefschetz map inpurely topological terms.

Author: ** Heath Emerson**

Source: https://arxiv.org/