Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index

TL;DR

This paper introduces a new invariant based on bordism techniques to analyze coincidences of fibre-preserving maps, connecting it to Dold's fixed point index and developing Nielsen and Reidemeister class concepts over a base space.

Contribution

It defines a novel bordism-based invariant for fibre-preserving maps, linking it to Dold's fixed point index and extending Nielsen coincidence theory over fibrations.

Findings

The invariant obstructs deforming pairs of fibre-preserving maps to be coincidence-free.

In special cases, the invariant equals Dold's fixed point index.

Computed minimal coincidence components for maps between S^1-bundles over S^1.

Abstract

Let M to B, N to B be fibrations and f1,f2 :M to N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f1,f2 over B to a coincidence free pair of maps.In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our {\omega}-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between S^1-bundles over S^1 as well as their Nielsen and Reidemeister numbers.