#### Geometry & Topology, Vol. 7 (2003)
Paper no. 16, pages 569--613.

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

### Wolfgang Lueck and Jonathan Rosenberg

**Abstract**.
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-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 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 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

**AMS subject classification**.
Primary: 19K33.
Secondary: 19K35, 19K56, 19L47, 58J22, 57R91, 57S30, 55P91.

**DOI:** 10.2140/gt.2003.7.569

**E-print:** `arXiv:math.KT/0208164`

Submitted to GT on 2 August 2002.
Paper accepted 9 October 2003.
Paper published 11 October 2003.

Notes on file formats
Wolfgang Lueck and Jonathan Rosenberg

Institut fur Mathematik und Informatik, Westfalische Wilhelms-Universtitat

Einsteinstr. 62, 48149 Munster, Germany

and

Department of Mathematics, University of Maryland

College Park, MD 20742, USA

Email: lueck@math.uni-muenster.de, jmr@math.umd.edu

URL: http://wwwmath.uni-muenster.de/u/lueck,
http://www.math.umd.edu/~jmr

GT home page

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