Self Coincidence Numbers and the Fundamental Group

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 will 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 , denoted Λf,f , equals zero. Since Λf,f 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.