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Structure and Ktheory of crossed products by proper actions, Expo
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Dualities in equivariant Kasparov theory
"... Abstract. We study several duality isomorphisms between equivariant bivariant Ktheory groups, generalising Kasparov’s first and second Poincaré duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised selfmaps. The second duality ..."
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Abstract. We study several duality isomorphisms between equivariant bivariant Ktheory groups, generalising Kasparov’s first and second Poincaré duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised selfmaps. 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 Gspace. This is a basic requirement for the dual Dirac method and allows us to describe the Baum–Connes assembly map via localisation of categories. Contents