Duality, Correspondences and the Lefschetz Map in Equivariant KK-Theory: A Survey

We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our results. We have organized this survey around the objective of computing a certain homotopy invariant of a space equipped with a (generally proper) action of a groupoid. This invariant is called the Lefschetz map. The Lefschetz map associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X. We will compute the Lefschetz map explicitly for a bundle of smooth manifolds over the base space of a proper groupoid, in which groupoid elements act by diffeomorphisms between fibres. We will use the topological model of equivariant KK-theory using correspondences originating in work of Paul Baum, Alain Connes and Georges Skandalis in the 1980's, and worked out fully in several of our recent articles, to do this. We include a discussion of this 'topological KK-theory' and the issue of when it agrees with the analytic theory. We also describe several other computations of the Lefschetz map: in the situation of a simplicial action of a group on a simplicial complex we compute the Lefschetz invariant of an equivariant cellular self-map, and in the situation of a smooth Riemannian manifold with isometric action of a group we compute the Lefschetz invariant of a smooth equivariant self-map satisfying a weak transversality condition.