Fixed Points and Coincidences in Torus Bundles

TL;DR

This paper develops a topological framework using normal bordism theory and Nielsen numbers to analyze fixed points and coincidences in torus bundles, providing explicit computations and criteria for fixed point free maps.

Contribution

It introduces a fiberwise approach to fixed point and coincidence problems in torus bundles, generalizing Nielsen theory with algebraic and geometric tools.

Findings

Determined minimum numbers of fixed points and coincidences for maps between torus bundles.

Provided algebraic descriptions using Reidemeister invariants and orbit analysis.

Established existence criteria for fixed point free fiberwise maps.

Abstract

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper we investigate fiberwise analoga and represent a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They allow also a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S1)2-bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.