Reidemeister coincidence invariants of fiberwise maps

TL;DR

This paper develops methods to compute Nielsen coincidence numbers for fiberwise maps between smooth fiber bundles, providing explicit calculations and criteria for deformability and fixed point properties, especially for torus bundles over the circle.

Contribution

It introduces techniques to calculate Reidemeister and Nielsen coincidence invariants for fiberwise maps, including explicit formulas and orbit set descriptions, advancing the understanding of coincidence theory in fiber bundle contexts.

Findings

Reidemeister set described as orbit set of a group operation of π1(B)

Minimum coincidence numbers determined for torus bundles over the circle

Odd order orbits are significant in the coincidence analysis

Abstract

Given two fiberwise maps f1, f2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of (f1,f2) as the orbit set of a group operation of {\pi}1(B). The size and number of orbits captures crucial extra information. E.g. for torus bundles of arbitrary dimensions over the circle this determines the minimum coincidence numbers of the pair (f1,f2) completely. In particular we can decide when f1 and f2 can be deformed away from one another or when a fiberwise selfmap can be made fixed point free by a suitable homotopy. In two concrete examples we calculate the minimum and Nielsen numbers for all pairs of fiberwise maps explicitly. Odd order orbits turn out to play a special role.