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

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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 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.