Selfcoincidences and roots in Nielsen theory

TL;DR

This paper investigates the conditions under which two maps from spheres to manifolds can be deformed to avoid coincidence points, revealing restrictions on Nielsen numbers and their implications for coincidence theory.

Contribution

It introduces new restrictions on Nielsen numbers in coincidence theory and analyzes the geometry of coincidence loci, especially in selfcoincidence and root cases.

Findings

Nielsen numbers can only be 0, 1, or the fundamental group's cardinality.

Most Nielsen classes are inessential in selfcoincidence cases.

Strong vanishing results for coincidence invariants.

Abstract

Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand the resulting bordism class of coincidence data and the corresponding Nielsen numbers are strong looseness obstructions. On the other hand the values which these invariants may possibly assume turn out to satisfy severe restrictions, e.g. the Nielsen numbers can only take the values 0, 1 or the cardinality of the fundamental group of N. In order to show this we compare different Nielsen classes in the root case (where f1 or f2 is constant) and we use the fact that all but possibly one Nielsen class are inessential in the selfcoincidence case (where f1 = f2). Also we deduce strong vanishing results.