Binomial rings, and integral homology of complements of compact toric arrangements
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arXiv
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| Format: | Preprint |
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2026
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| author | Gorinov, Alexey G. Zakharov, Alexander V. |
| author_facet | Gorinov, Alexey G. Zakharov, Alexander V. |
| contents | An \emph{affine subtorus} of the compact torus $T=(S^1)^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\ldots, T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z}_{(p)})$. %The complex is determined by the integral homology maps induced by the inclusions $T_J\subset T_I$ where $I\subset J\subset\{1,\ldots, k\}$ and $T_I$ denotes $\bigcap_{i\in I} T_i$. Our main tool is the binomial models for spaces constructed by T.~Ekedahl. We use these results to express the groups $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z})$. We also show that the Mayer-Vietoris spectral sequence that converges to the homology of $T-\bigcup_{i=1}^k T_i$ collapses at the second page rationally, and also integrally under some assumptions on the arrangement $T_1,\ldots, T_k$, with all extension problems being trivial in the latter case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07902 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Binomial rings, and integral homology of complements of compact toric arrangements Gorinov, Alexey G. Zakharov, Alexander V. Algebraic Topology Category Theory An \emph{affine subtorus} of the compact torus $T=(S^1)^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\ldots, T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z}_{(p)})$. %The complex is determined by the integral homology maps induced by the inclusions $T_J\subset T_I$ where $I\subset J\subset\{1,\ldots, k\}$ and $T_I$ denotes $\bigcap_{i\in I} T_i$. Our main tool is the binomial models for spaces constructed by T.~Ekedahl. We use these results to express the groups $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z})$. We also show that the Mayer-Vietoris spectral sequence that converges to the homology of $T-\bigcup_{i=1}^k T_i$ collapses at the second page rationally, and also integrally under some assumptions on the arrangement $T_1,\ldots, T_k$, with all extension problems being trivial in the latter case. |
| title | Binomial rings, and integral homology of complements of compact toric arrangements |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2601.07902 |