Dualities in equivariant Kasparov theory

We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov’s first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Connes assembly map via localisation of categories.

💡 Research Summary

The paper develops two fundamental duality isomorphisms within equivariant Kasparov theory (KK^G) and demonstrates how they unify several central topics in non‑commutative geometry. The first duality generalises Kasparov’s original Poincaré duality to the setting of a proper G‑space X equipped with a G‑equivariant K‑orientation. By constructing a Dirac class D_X∈KK^G(C_0(X),ℂ) and a Thom class Θ_X∈KK^G(ℂ,C_0(X)), the authors obtain a natural isomorphism
 KK^G(A,B) ≅ KK^G(A⊗C_0(X), B⊗C_0(X)).
This machinery is then used to define an equivariant Lefschetz invariant for any G‑equivariant self‑map f:X→X:
 Lef_G(f)=Θ_X⊗_{C_0(X)}