Nielsen-Borsuk-Ulam number for maps between tori

TL;DR

This paper calculates the Nielsen-Borsuk-Ulam number for selfmaps and free involutions on tori up to dimension three, establishing these tori as Wecken spaces in this context, which informs about minimal coincidence pairs.

Contribution

It provides explicit computations of the Nielsen-Borsuk-Ulam number for tori and proves that low-dimensional tori are Wecken spaces in this theory.

Findings

Computed Nielsen-Borsuk-Ulam numbers for tori up to dimension three.

Established that tori $ ext{T}^1$, $ ext{T}^2$, and $ ext{T}^3$ are Wecken spaces.

Determined lower bounds for minimal coincidence pairs in homotopy classes.

Abstract

We compute the Nielsen-Borsuk-Ulam number for any selfmap of $n-$torus, $\mathbb{T}^n$, as well as any free involution $\tau$ in $\mathbb{T}^n$, with $n \leqslant 3$. Finally, we conclude that the tori, $\mathbb{T}^1$, $\mathbb{T}^2$ and $\mathbb{T}^3$, are Wecken spaces in Nielsen-Borsuk-Ulam theory. Such a number is a lower bound for the minimal number of pair of points such that $f(x)=f(\tau(x))$ in a given homotopy class of maps.