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

The degree of a map between orientable manifolds is a fundamental concept in topology that aids in understanding the structure and properties of the manifolds and the maps between them. Numerous studies have been conducted on the degree of maps between orientable topological spaces. For each $d\in \mathbb{Z}$, we construct a degree $d$ simplicial map from a $(2(n+1)\max \{\mathrm{\mid }d\mathrm{\mid },1\})$-facet colored triangulation of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$ to the standard $2(n+1)$-facet colored triangulation of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. We demonstrate that these are the minimal possible colored triangulations for a degree $d$ simplicial self-map of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$, where $n\ge 2$. Additionally, we construct a minimal degree $d$ simplicial map from a closed orientable $n$-manifold to ${\mathbb{S}}^n$, where $n\ge 1$.