A construction of homotopically non-trivial embedded spheres for hyperplane arrangements
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911894517841920 |
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| author | Yoshinaga, Masahiko |
| author_facet | Yoshinaga, Masahiko |
| contents | We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the half-spaces. We then prove that the sphere is homotopically trivial if and only if the system of half-spaces is globally consistent. To prove its non-triviality, we compute the twisted intersection number of the sphere with a specific, explicitly constructed twisted Borel-Moore cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20010 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A construction of homotopically non-trivial embedded spheres for hyperplane arrangements Yoshinaga, Masahiko Geometric Topology Algebraic Topology Combinatorics We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the half-spaces. We then prove that the sphere is homotopically trivial if and only if the system of half-spaces is globally consistent. To prove its non-triviality, we compute the twisted intersection number of the sphere with a specific, explicitly constructed twisted Borel-Moore cycle. |
| title | A construction of homotopically non-trivial embedded spheres for hyperplane arrangements |
| topic | Geometric Topology Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2405.20010 |