On vertex-minimal simplicial maps to the sphere

Abstract

For positive integers $n,d$, let $\lambda (n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor \frac{n+1}{2}\rfloor$, the function $\lambda (n,d{)}^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in ${\mathbb{R}}^{n+1}$.

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