  For M and N closed oriented connected smooth manifolds of the same dimension,
we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M-->
N.We show that the evaluation map from the space of maps to the manifold N
induces a nontrivial homomorphism on the fundamental group only if the self
coincidence number of f equals zero. Since the self intersection number is
equal to the product of the degree of f and the Euler--Poincare number of N, we
obtain results related to earlier results about the evaluation map and the
Euler--Poincare number.
