Minimal simplicial degree $d$ self-maps of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$

Abstract

The degree of a map between orientable manifolds is a fundamental concept in topology, providing important information about the structure of manifolds and the behavior of maps between them. A simplicial cell complex $K$ is called a \emph{colored triangulation} of a closed PL $n$-manifold $M$ if the $1$-skeleton of $K$ admits a proper vertex-coloring with $n+1$ colors and $|K|$ is PL-homeomorphic to $M$. In this article, we construct, for every $d\in \mathbb{Z}$ and $n\ge 2$, a degree $d$ simplicial map from a $(2(n+1)\max \{|d|,1\})$-facet colored triangulation of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$ to the standard $2(n+1)$-facet colored triangulation of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. Additionally, for every $d\in \mathbb{Z}$ and $n\ge 2$, we construct a degree $d$ simplicial map from a $(2\max \{|d|,1\})$-facet colored triangulation of ${\mathbb{S}}^n$ to the standard $2$-facet colored triangulation of ${\mathbb{S}}^n$. For $M={\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$ and ${\mathbb{S}}^n$, with $n\ge 2$, these simplicial degree $d$ self-maps of $M$ are minimal with respect to their standard colored triangulations, in the sense that there does not exist a colored triangulation $K$ of $M$ with fewer facets than the constructed one that admits a simplicial map $f:K\to K^{\prime }$ of degree $d$, where $K^{\prime }$ denotes the standard colored triangulation of $M$.

14 Pages, 7 figures. To appear in Bulletin of the Belgian Mathematical Society - Simon Stevin