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 $$f : M \longrightarrow N$$ . 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 $$\Lambda_{f,f}$$ , equals zero. Since $$\Lambda_{f,f}$$ is equal to the product of the degree of f and the Euler characteristic of N, we obtain results related to earlier results about the evaluation map and the Euler characteristic.