Duality, Correspondences and the Lefschetz Map in Equivariant Kk-theory: a Survey

We survey work by the author and Ralf Meyer on equivariant KKtheory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The latter associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X admitting a dual. We want to describe it explicitly in the setting of bundles of smooth manifolds over the base space of a groupoid, in which groupoid elements act by diffeomorphisms between fibres. To get the required description we develop a topological model of equivariant KK-theory by way of a theory of correspondences, building on ideas of Paul Baum, Alain Connes and Georges Skandalis in the 1980’s. This model agrees with the analytic model for bundles of smooth manifolds under some technical conditions related to the existence of equivariant vector bundles. Subject to these conditions we obtain a computation of the Lefschetz map in purely topological terms. We finish by stating a generalisation of the classical Lefschetz fixed-point theorem that applies to correspondences, not just maps. The papers [13], [11], [12], [16] present a study of the equivariant Kasparov groups KK ( C0(X), C0(Y ) ) where G is a locally compact Hausdorff groupoid with Haar system and X and Y are G-spaces, usually with X a proper G-space. This program builds on work of Kasparov, Connes and Skandalis done mainly in the 1980’s. At that point, the main interest was the index theorem of Atiyah and Singer and its generalisations, and later, the Dirac dual-Dirac method and the Novikov conjecture. For our part, we are motivated by the idea of developing Euler characteristics and Lefschetz formulas in equivariant KK-theory. Via the Baum-Connes isomorphism – when it applies – this contributes to noncommutative topology and index theory. Our program started in [13] where we found the Lefschetz map in connection with a K-theory problem. We will give the definition of the Lefschetz map in the first section, but for now record that it has the form (0.1) Lef : KK ∗ ( C0(X ×Z X), C0(X) ) → KK∗ (C0(X), C0(Z) ) , where we always denote by Z the base space of the groupoid. This map is defined under certain somewhat technical circumstances, but, again, these normally involve proper G-spaces X . The domain of the Lefschetz map is very closely related to the simpler-looking group KK∗ ( C0(X), C0(X) ) : the latter group maps in a natural way to the domain in (0.1) and this map is an isomorphism when the anchor map X → Z is a proper map. This means that the Lefschetz map can be used to assign an invariant, which is an equivariant K-homology class, to an equivariant Kasparov self-morphism of X . We call this class the Lefschetz invariant of the map. It bears consideration even when G is the trivial groupoid, and the reader can do worse than to consider this case to begin with, although by doing so one misses the applications to noncommutative topology. 2000 Mathematics Subject Classification. 19K35, 46L80. Heath Emerson was supported by a National Science and Engineering Council of Canada (NSERC) Discovery grant.