Minimum numbers and Wecken theorems in topological coincidence theory. I

TL;DR

This paper investigates the relationship between minimum numbers and Nielsen numbers in topological coincidence theory, providing tools to determine when they coincide and exploring their geometric implications across various manifolds.

Contribution

It introduces new criteria and methods for comparing minimum numbers and Nielsen numbers, especially in higher codimension cases, with applications to diverse geometric settings.

Findings

Criteria for when minimum numbers equal Nielsen numbers

Development of tools for Nielsen number computation

Applications to spheres, projective spaces, and Lie groups

Abstract

Minimum numbers measure the obstruction to removing coincidences of two given maps (between smooth manifolds M and N of dimensions m and n, resp.). In this paper we compare them to four distinct types of Nielsen numbers. These agree with the classical Nielsen number when m = n (e.g. in the fixed point setting where M = N and one of the maps is the identity map). However, in higher codimensions m - n > 0 their definitions and computations involve distinct aspects of differential topology and homotopy theory. We develop tools which help us 1.) to decide when a minimum number is equal to a Nielsen number ("Wecken theorem"), and 2.) to determine Nielsen numbers. Here certain homotopy theoretical criteria play a central role. E.g. failures of the "Wecken condition" (cf. definition 1.18 below) can have very interesting geometric consequences. The selfcoincidence case where the two maps are homotopic turns out to be particularly illuminating. We give many concrete applications in special settings where M or N are spheres, spherical space forms, projective spaces, tori, Stiefel manifolds or Grassmannians. Already in the simplest examples an important role is played e.g. by Kervaire invariants, all versions of Hopf invariants (a la James, Hilton, Ganea,...) and the elements in the stable homotopy of spheres defined by invariantly framed Lie groups.