Nielsen-Borsuk-Ulam number for maps between tori

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 $n⩽3$. 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.

arXiv comment21 pages