  Using Poincar\'e duality in K-theory, we state and prove a Lefschetz fixed
point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$
coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and
$X\cross G$ cocompact and proper.
  The formula in question expresses the graded trace of the map on rationalized
K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the
Lefschetz number, in terms of fixed orbits and representation-theoretic data
connected with certain isotropy subgroups of the isotropy group at that point.
The technique is to use noncommutative Poinca\'e duality and the formal
Lefschetz lemma of the second author.
