Face numbers of triangulations of manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914646893527040 |
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| author | Spreer, Jonathan Tobin, Lucy |
| author_facet | Spreer, Jonathan Tobin, Lucy |
| contents | In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$ of a simply connected $4$-manifold $\mathcal{M}$ with $n$ pentachora, an upper bound on the number of vertices $v$ of $\mathcal{T}$ as a function of $n$ yields a lower bound for $n$ depending only on the second Betti number $β_2(\mathcal{M})$ of $\mathcal{M}$. Within this framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq 2β_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured bound is shown to be almost tight for all values of $β_2(\mathcal{M})$, with a gap of at most two.
We extend our conjecture to arbitrary dimensions and show that an $n$-facet triangulation of an odd-dimensional $d$-manifold, $n \geq d$, can have at most $n + \frac{d-1}{2}$ vertices, and conjecture that, for $d$ even, the bound is $\frac{n}{2}+d$. We show that these (conjectured) bounds are (would be) tight for all odd (even) dimensions and all values of $n \geq d$. Finally, we give necessary conditions for the dual graph of $\mathcal{T}$ to satisfy our conjecture. We furthermore present families of $4$-dimensional pseudomanifolds with singularities in their edge links that have more than $\frac{n}{2}+4$ vertices, thereby proving that the manifold condition is necessary for our conjecture to hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Face numbers of triangulations of manifolds Spreer, Jonathan Tobin, Lucy Geometric Topology Combinatorics 57Q15, 57Q05, 57K40 In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$ of a simply connected $4$-manifold $\mathcal{M}$ with $n$ pentachora, an upper bound on the number of vertices $v$ of $\mathcal{T}$ as a function of $n$ yields a lower bound for $n$ depending only on the second Betti number $β_2(\mathcal{M})$ of $\mathcal{M}$. Within this framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq 2β_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured bound is shown to be almost tight for all values of $β_2(\mathcal{M})$, with a gap of at most two. We extend our conjecture to arbitrary dimensions and show that an $n$-facet triangulation of an odd-dimensional $d$-manifold, $n \geq d$, can have at most $n + \frac{d-1}{2}$ vertices, and conjecture that, for $d$ even, the bound is $\frac{n}{2}+d$. We show that these (conjectured) bounds are (would be) tight for all odd (even) dimensions and all values of $n \geq d$. Finally, we give necessary conditions for the dual graph of $\mathcal{T}$ to satisfy our conjecture. We furthermore present families of $4$-dimensional pseudomanifolds with singularities in their edge links that have more than $\frac{n}{2}+4$ vertices, thereby proving that the manifold condition is necessary for our conjecture to hold. |
| title | Face numbers of triangulations of manifolds |
| topic | Geometric Topology Combinatorics 57Q15, 57Q05, 57K40 |
| url | https://arxiv.org/abs/2401.11152 |