Axioms for the coincidence index of maps between manifolds of the same dimension

TL;DR

This paper develops axiomatic characterizations for the local coincidence index and Reidemeister trace for maps between manifolds of the same dimension, extending classical results to more general settings.

Contribution

It introduces axioms that uniquely determine the local index and Reidemeister trace for coincidence theory between manifolds, including non-orientable cases and arbitrary pairs of maps.

Findings

Axioms characterize the local index as an integer or in Z⊕Z₂.

The group of index values is determined by the axioms.

Axioms for the local Reidemeister trace extend known results to broader contexts.

Abstract

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in $\Z\oplus \Z_2$. We also show in each setting that the group of values for the index (either $\Z$ or $\Z\oplus \Z_2$) is determined by the axioms. Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which charaterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds.