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Equivariant Euler characteristics and $K$–homology Euler classes for proper cocompact $G$–manifolds

Wolfgang Lueck and Jonathan Rosenberg

Geometry & Topology 7 (2003) 569–613

arXiv: math.KT/0208164


Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO–homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from UG(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no “higher” equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L–fixed point sets ML, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S3 on the 3–sphere for which the equivariant Euler class has order 2, so there is also some torsion information.

equivariant $K$–homology, de Rham operator, signature operator, Kasparov theory, equivariant Euler characteristic, fixed sets, cyclic subgroups, Burnside ring, Euler operator, equivariant Euler class, universal equivariant Euler characteristic
Mathematical Subject Classification 2000
Primary: 19K33
Secondary: 19K35, 19K56, 19L47, 58J22, 57R91, 57S30, 55P91
Forward citations
Received: 2 August 2002
Accepted: 9 October 2003
Published: 11 October 2003
Proposed: Steve Ferry
Seconded: Martin Bridson, Frances Kirwan
Wolfgang Lueck
Institut für Mathematik und Informatik
Westfälische Wilhelms-Universtität
Einsteinstr. 62
48149 Münster
Jonathan Rosenberg
Department of Mathematics
University of Maryland
College Park
Maryland 20742