  Suppose one is given a discrete group G, a cocompact proper G-manifold M, and
a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f,
which is globally defined in terms of cellular chain complexes, and the local
equivariant Lefschetz class of f, which is locally defined in terms of fixed
point data. We prove the equivariant Lefschetz fixed point theorem, which says
that these two classes agree. As a special case, we prove an equivariant
Poincare-Hopf Theorem, computing the universal equivariant Euler characteristic
in terms of the zeros of an equivariant vector field, and also obtain an
orbifold Lefschetz fixed point theorem. Finally, we prove a realization theorem
for universal equivariant Euler characteristics.
