Simplicial spheres with $g_k=1$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918290298765312 |
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| author | Novik, Isabella Zheng, Hailun |
| author_facet | Novik, Isabella Zheng, Hailun |
| contents | For $d\geq 4$, Kalai (1987) characterized all simplicial $(d-1)$-spheres with $g_2=0$, and for $k\geq 2$ and $d\geq 2k$, Murai and Nevo (2013) characterized all simplicial $(d-1)$-spheres with $g_k=0$. In addition, for $d\geq 4$, Nevo and Novinsky (2011) characterized all simplicial $(d-1)$-spheres with $g_2=1$. Motivated by these results, we characterize, for any $k\geq 2$ and $d\geq 2k+1$, all simplicial $(d-1)$-spheres with no missing faces of dimension larger than $d-k$ that satisfy $g_k=1$. When $d=2k$, we obtain a characterization of simplicial $(d-1)$-spheres with $g_k=1$ and no missing faces of dimension greater than $k$, under the additional assumption that there exists at least one missing face of dimension $k$. Finally, for $k=3$, we are able to remove this assumption and characterize all simplicial $5$-spheres with no missing faces of dimension larger than $3$ that satisfy $g_3=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplicial spheres with $g_k=1$ Novik, Isabella Zheng, Hailun Combinatorics 13F55, 52B05, 05E45 For $d\geq 4$, Kalai (1987) characterized all simplicial $(d-1)$-spheres with $g_2=0$, and for $k\geq 2$ and $d\geq 2k$, Murai and Nevo (2013) characterized all simplicial $(d-1)$-spheres with $g_k=0$. In addition, for $d\geq 4$, Nevo and Novinsky (2011) characterized all simplicial $(d-1)$-spheres with $g_2=1$. Motivated by these results, we characterize, for any $k\geq 2$ and $d\geq 2k+1$, all simplicial $(d-1)$-spheres with no missing faces of dimension larger than $d-k$ that satisfy $g_k=1$. When $d=2k$, we obtain a characterization of simplicial $(d-1)$-spheres with $g_k=1$ and no missing faces of dimension greater than $k$, under the additional assumption that there exists at least one missing face of dimension $k$. Finally, for $k=3$, we are able to remove this assumption and characterize all simplicial $5$-spheres with no missing faces of dimension larger than $3$ that satisfy $g_3=1$. |
| title | Simplicial spheres with $g_k=1$ |
| topic | Combinatorics 13F55, 52B05, 05E45 |
| url | https://arxiv.org/abs/2601.10072 |