Cleanliness and the Varchenko-Gelfand algebra
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917355994480640 |
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| author | Denham, Graham Dorpalen-Barry, Galen Proudfoot, Nicholas |
| author_facet | Denham, Graham Dorpalen-Barry, Galen Proudfoot, Nicholas |
| contents | A central question in the theory of hyperplane arrangements is when the complement of a complex arrangement is aspherical. Barkley and Speyer introduced a class of real arrangements that are called "clean," and Yoshinaga proved that every real arrangement whose complexification is $K(π,1)$ is clean. We show that cleanliness is equivalent to a natural statement about the Varchenko-Gelfand ring, which in practice allows for fast calculation. We conclude with an investigation of the relationships between various properties of arrangements, including cleanliness and the asphericity of the arrangement complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cleanliness and the Varchenko-Gelfand algebra Denham, Graham Dorpalen-Barry, Galen Proudfoot, Nicholas Combinatorics 52C35, 52C40 A central question in the theory of hyperplane arrangements is when the complement of a complex arrangement is aspherical. Barkley and Speyer introduced a class of real arrangements that are called "clean," and Yoshinaga proved that every real arrangement whose complexification is $K(π,1)$ is clean. We show that cleanliness is equivalent to a natural statement about the Varchenko-Gelfand ring, which in practice allows for fast calculation. We conclude with an investigation of the relationships between various properties of arrangements, including cleanliness and the asphericity of the arrangement complement. |
| title | Cleanliness and the Varchenko-Gelfand algebra |
| topic | Combinatorics 52C35, 52C40 |
| url | https://arxiv.org/abs/2512.10077 |