New York Journal of Mathematics
Volume 16 (2010) 245-313


Heath Emerson and Ralf Meyer

Dualities in equivariant Kasparov theory

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Published: October 21, 2010
Keywords: K-theory, K-homology, duality, groupoids, Euler characteristics
Subject: 19K35, 46L80

We study several duality isomorphisms between equivariant bivariant K-theory 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 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.


Heath Emerson was supported by a National Science and Research Council of Canada Discovery grant. Ralf Meyer was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.

Author information

Heath Emerson:
Department of Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C., Canada V8W 3P4

Ralf Meyer:
Mathematisches Institut and Courant Research Centre "Higher Order Structures'', Georg-August Universität Göttingen, Bunsenstra{\ss}e 3-5, 37073 Göttingen, Germany