On vertex-minimal simplicial maps to the sphere
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915646696062976 |
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| author | Ryabichev, Andrey |
| author_facet | Ryabichev, Andrey |
| contents | 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 $\mathbb{R}^{n+1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On vertex-minimal simplicial maps to the sphere Ryabichev, Andrey Combinatorics Geometric Topology 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 $\mathbb{R}^{n+1}$. |
| title | On vertex-minimal simplicial maps to the sphere |
| topic | Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2512.01137 |