On vertex-minimal simplicial maps to the sphere

TL;DR

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Contribution

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Abstract

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that $\lim_{d\to\infty}\frac{\lambda(n,d)}d=0$ for any $n\ge3$, disproving O. Musin's conjecture. Using similar idea, for any $C$ we construct a triangulation of $\mathbb{S}^n$, $n\ge3$, for which $\frac{f_j}{f_i}>C$, for any $0\le i<j\le n$ such that $i<\lfloor\frac{n-1}2\rfloor$. All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.