#### Volume 7, issue 2 (2003)

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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
##### 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 $KO$–homology of $M$. The universal equivariant Euler characteristic of $M$, which lives in a group ${U}^{G}\left(M\right)$, counts the equivariant cells of $M$, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from ${U}^{G}\left(M\right)$ 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
##### Mathematical Subject Classification 2000
Primary: 19K33
Secondary: 19K35, 19K56, 19L47, 58J22, 57R91, 57S30, 55P91