Cubulating the sphere with many facets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908280093147136 |
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| author | Avvakumov, Sergey Hubard, Alfredo |
| author_facet | Avvakumov, Sergey Hubard, Alfredo |
| contents | For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $Ω(n^{5/4})$.
This disproves a conjecture of Kalai's stating that the number of faces (of all dimensions) of cubical spheres is maximized by the boundaries of neighbourly cubical polytopes. The conjecture was already known to be false for $d=3$, $n=64$. Our construction disproves it for all $d\geq 3$ and $n$ sufficiently large. Moreover, since neighborly cubical polytopes have roughly $n (\log n)^{d/2}$ facets, we show that even the order of growth (at least for the number of facets) in the conjecture is wrong. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cubulating the sphere with many facets Avvakumov, Sergey Hubard, Alfredo Combinatorics Geometric Topology For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $Ω(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of faces (of all dimensions) of cubical spheres is maximized by the boundaries of neighbourly cubical polytopes. The conjecture was already known to be false for $d=3$, $n=64$. Our construction disproves it for all $d\geq 3$ and $n$ sufficiently large. Moreover, since neighborly cubical polytopes have roughly $n (\log n)^{d/2}$ facets, we show that even the order of growth (at least for the number of facets) in the conjecture is wrong. |
| title | Cubulating the sphere with many facets |
| topic | Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2503.18047 |