  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.
