Toric arrangements and Bloch-Kato pro-$p$ groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Delucchi, Emanuele Marmo, Ettore |
| author_facet | Delucchi, Emanuele Marmo, Ettore |
| contents | We prove a purely combinatorial obstruction for the Bloch-Kato property within the class of fundamental groups of complement manifolds of toric arrangements (i.e., arrangements of hypersurfaces in the complex torus). As a stepping stone we obtain a combinatorial obstruction for the cohomology of a supersolvable arrangement to be generated in degree 1. Our result allows us to prove that - for all prime numbers $p$, the pro-$p$ completion of the pure braid group on $k$ strands has the Bloch-Kato property if and only if $k\leq 3$; - for all prime numbers $p$, the pro-$p$ completion of the pure mapping class group of the sphere $S^2$ with $k$ punctures has the Bloch-Kato property if and only if $k\leq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toric arrangements and Bloch-Kato pro-$p$ groups Delucchi, Emanuele Marmo, Ettore Group Theory Combinatorics Primary: 52C35, 20E18, Secondary: 06A07, 20F36, 12F10 We prove a purely combinatorial obstruction for the Bloch-Kato property within the class of fundamental groups of complement manifolds of toric arrangements (i.e., arrangements of hypersurfaces in the complex torus). As a stepping stone we obtain a combinatorial obstruction for the cohomology of a supersolvable arrangement to be generated in degree 1. Our result allows us to prove that - for all prime numbers $p$, the pro-$p$ completion of the pure braid group on $k$ strands has the Bloch-Kato property if and only if $k\leq 3$; - for all prime numbers $p$, the pro-$p$ completion of the pure mapping class group of the sphere $S^2$ with $k$ punctures has the Bloch-Kato property if and only if $k\leq 4$. |
| title | Toric arrangements and Bloch-Kato pro-$p$ groups |
| topic | Group Theory Combinatorics Primary: 52C35, 20E18, Secondary: 06A07, 20F36, 12F10 |
| url | https://arxiv.org/abs/2507.16428 |