Nielsen coincidence numbers, Hopf invariants and spherical space forms

Abstract

Given two maps between smooth manifolds, the obstruction to removing their coincidences (via homotopies) is measured by minimum numbers. In order to determine them we introduce and study an infinite hierarchy of Nielsen numbers N_i, i = 0, 1, ..., \infty. They approximate the minimum numbers from below with decreasing accuracy, but they are (in principle) more easily computable as i grows. If the domain and the target manifold have the same dimension (e.g. in the fixed point setting) all these Nielsen numbers agree with the classical definition. However, in general they can be quite distinct. While our approach is very geometric the computations use the techniques of homotopy theory and, in particular, all versions of Hopf invariants (a la Ganea, Hilton, James..). As an illustration we determine all Nielsen numbers and minimum numbers for pairs of maps from spheres to spherical space forms. Maps into even dimensional real projective spaces turn out to produce particularly interesting coincidence phenomena.