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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.

Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey