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 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 Lefschetz map 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 proper groupoid, in which groupoid elements act by diffeomorphisms between fibres. To get the required description we describe 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 that appeared 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.

💡 Research Summary

The paper is a comprehensive survey of the author’s joint work with Ralf Meyer on equivariant KK‑theory (KK^G) and its applications to a homotopy‑invariant called the Lefschetz map. The central theme is the use of duality: a G‑space X is said to admit a dual D if there exist elements Δ∈KK^G(X×D, X) and Δ̂∈KK^G(X, X×D) that satisfy the usual Poincaré‑duality relations. When such a dual exists, any equivariant Kasparov self‑morphism f∈KK^G(X, X) can be “paired” with the dual to produce a class
 L(f) = Δ̂ ⊗_D f ⊗D Δ ∈ K^G*(X).
This construction generalizes the classical Lefschetz number, which counts fixed points of a map, to a K‑homology class that encodes fixed‑point data in a non‑commutative, equivariant setting.

To make L(f) computable, the authors develop a topological model of equivariant KK‑theory based on correspondences, a notion that goes back to Baum, Connes, and Skandalis. A correspondence from X to Y is a triple (M, E, φ) where M is a G‑equivariant smooth manifold equipped with proper maps to X and Y, E is a G‑equivariant complex vector bundle over M, and φ is a K‑theory class on M. The key insight is that, for bundles of smooth manifolds over the base of a proper groupoid G, every Kasparov class can be represented by such a correspondence, provided certain technical hypotheses hold. These hypotheses amount to the existence of enough G‑equivariant vector bundles on the fibres (often guaranteed when the fibres have sufficiently large dimension or admit a G‑invariant Spin^c structure).

The paper proceeds in several steps:

1. Duality Framework – The authors formalize the notion of a dual D for a G‑space X, prove the existence of the “diagonal” elements Δ and Δ̂, and show how they give rise to the Lefschetz map L: KK^G(X, X) → K^G_*(X). They also discuss functorial properties, such as naturality with respect to G‑equivariant maps and compatibility with external products.

2. Correspondence Model – Building on the Baum‑Connes‑Skandalis picture, they define a category Corr^G whose morphisms are correspondences. They prove that, under the vector‑bundle hypothesis, the analytic KK‑group KK^G(X, Y) is naturally isomorphic to the homotopy classes of correspondences Corr^G(X, Y). This is achieved by constructing explicit “assembly” and “descent” maps and showing they are inverses up to homotopy.

3. Equivariant Vector Bundles and Technical Conditions – The authors analyze when the required equivariant vector bundles exist. They give sufficient conditions, such as properness of the groupoid action, existence of a G‑invariant Riemannian metric on the fibres, and the presence of a G‑equivariant Spin^c structure. They also discuss how to handle cases where these conditions fail, e.g., by passing to a Morita‑equivalent groupoid or by stabilizing the bundles.

4. Computation of the Lefschetz Map – With the correspondence model in hand, the Lefschetz map becomes a purely topological construction. Given f∈KK^G(X, X), one first replaces f by a correspondence (M_f, E_f, φ_f). The fixed‑point set of the correspondence, M_f^G, is a G‑invariant submanifold of M_f. There is a transfer (or “wrong‑way”) map τ: K^G_(M_f^G) → K^G_(X) induced by the proper map M_f^G → X. The Lefschetz class is then
    L(f) = τ(