On vertex-minimal simplicial maps to the sphere
Andrey Ryabichev
TL;DR
The paper studies minimal vertex counts λ(n,d) for triangulated S^n admitting degree-d simplicial maps to the boundary of an (n+1)-simplex, proving that for n≥3 the ratio λ(n,d)/d → 0, thus refuting Musin's conjecture. It achieves this with a join-based construction that yields high-degree maps with only linearly many vertices, aided by a subdivision method to adjust degree by ±1, and by leveraging multiplicativity of degree under joins. Additionally, the authors construct convex-polytope boundary triangulations with highly unbalanced f-vectors, showing f_j/f_i > C for certain indices, which demonstrates strong growth in certain face counts while preserving polytope realizability. The work advances understanding of asymptotic vertex requirements for degree maps and raises questions about exact values and degeneracy constraints on simplicial maps.
Abstract
For positive integers $n,d$, let $λ(n,d)$ 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 $\lim_{d\to\infty}\frac{λ(n,d)}d=0$ 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 $\mathbb{R}^{n+1}$.