Self coincidence numbers and the fundamental group

Abstract

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.