Nielsen coincidence numbers, Hopf invariants and spherical space forms

TL;DR

This paper introduces an infinite hierarchy of Nielsen numbers to measure the minimal coincidences of maps between manifolds, providing new computable invariants that generalize classical fixed point theory and are applied to spherical space forms.

Contribution

The paper develops a novel hierarchy of Nielsen numbers that approximate minimal coincidence numbers and connects them with Hopf invariants, extending classical fixed point results to more general manifolds.

Findings

All Nielsen numbers and minimum numbers are computed for maps from spheres to spherical space forms.

Distinct Nielsen numbers can occur in general, unlike the classical case where they coincide.

Maps into even-dimensional real projective spaces exhibit interesting coincidence phenomena.

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.