Minimal simplicial spherical mappings with a given degree

TL;DR

This paper investigates the minimal number of vertices needed in a triangulation of an n-sphere to map simplicially to a simplex boundary with a specified degree, providing bounds, formulas, and reduction techniques.

Contribution

It establishes upper bounds for vertex counts in higher dimensions and exact formulas for small degrees, introducing a key identity to reduce complex cases to simpler ones.

Findings

mbda(n,d) = n+d+3 for n geq 3 and d=2,3,4

A reduction identity mbda(n,d) = mbda(d-1,d) + n - d + 1

Constructive methods for triangulation modifications

Abstract

This paper studies the minimal number of vertices $\lambda(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for $\lambda(n,d)$ in dimensions $n \geq 3$. Furthermore, we provide exact formulas for small values of $d$, showing that $\lambda(n,d)=n+d+3$ for $n \geq 3$ and $d=2,3,4$. A key technical result is the identity $\lambda(n,d) = \lambda(d-1,d) + n - d + 1$ for $n \geq d$, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments.