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

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 $K O$ –homology of $M$ . The universal equivariant Euler characteristic of $M$ , which lives in a group $U^{G} (M)$ , counts the equivariant cells of $M$ , taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from $U^{G} (M)$ to the equivariant $K O$ -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 $M^{L}$ , where $L$ runs through the finite cyclic subgroups of $G$ . However, we give an example of an action of the symmetric group $S_{3}$ on the 3–sphere for which the equivariant Euler class has order 2, so there is also some torsion information.

Keywords

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

References

Forward citations

Publication

Received: 2 August 2002

Accepted: 9 October 2003

Published: 11 October 2003

Proposed: Steve Ferry

Seconded: Martin Bridson, Frances Kirwan

Authors

Wolfgang Lueck
Institut für Mathematik und Informatik Westfälische Wilhelms-Universtität Einsteinstr. 62 48149 Münster Germany

Jonathan Rosenberg
Department of Mathematics University of Maryland College Park Maryland 20742 USA

Volume 7, issue 2 (2003)

Wolfgang Lueck and Jonathan Rosenberg