Dualities in equivariant Kasparov theory

We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov’s first and second Poincar´e 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.