Let be a countable discrete
group and let be a smooth
proper cocompact -manifold
without boundary. The Euler operator defines via Kasparov theory
an element, called the equivariant Euler class, in the equivariant
–homology
of .
The universal equivariant Euler characteristic of
, which lives in a
group , counts the
equivariant cells of ,
taking the component structure of the various fixed point sets
into account. We construct a natural homomorphism from
to the equivariant
-homology
of .
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
–fixed point
sets , where
runs through the finite
cyclic subgroups of .
However, we give an example of an action of the symmetric group
on
the 3–sphere for which the equivariant Euler class has order 2, so there is also some
torsion information.