Nielsen numbers in topological coincidence theory

TL;DR

This paper explores Nielsen numbers in topological coincidence theory, comparing various invariants, computing them for specific map classes, and establishing a Wecken theorem, with corrections to prior quaternionic cases.

Contribution

It introduces and compares four Nielsen numbers in coincidence theory, computes them for maps between spheres and projective spaces, and proves a Wecken theorem in this context.

Findings

All four Nielsen numbers coincide with classical fixed point Nielsen number in fixed point setting.

Explicit calculations of Nielsen numbers for maps from spheres to projective spaces.

Counterexamples to Nielsen number equality when dimension conditions are not met.

Abstract

We discuss coincidences of pairs (f_1, f_2) of maps between manifolds. We recall briefly the definition of four types of Nielsen numbers which arise naturally from the geometry of generic coincidences. They are lower bounds for the minimum numbers MCC and MC which measure to some extend the 'essential' size of a coincidence phenomenon. In the setting of fixed point theory these Nielsen numbers all coincide with the classical notion but in general they are distinct invariants. We illustrate this by many examples involving maps from spheres to the real, complex or quaternionic projective space KP(n'). In particular, when n' is odd and K = R or C or when n' = 23 mod 24 and K = H, we compute the minimum number MCC and all four Nielsen numbers for every pair of these maps, and we establish a 'Wecken theorem' in this context (in the process we correct also a mistake in previous work concerning the quaternionic case). However, when n' is even, counterexamples can occur, detected e.g. by Kervaire invariants.