Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911085092667392 |
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| author | Sadovek, Nikola Soberón, Pablo |
| author_facet | Sadovek, Nikola Soberón, Pablo |
| contents | The Tverberg--Vrećica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg--Vrećica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Soberón, and provide a direct proof of the complex central transversal theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems Sadovek, Nikola Soberón, Pablo Combinatorics Algebraic Topology 52A35, 55N91, 55R91 The Tverberg--Vrećica conjecture claims a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg--Vrećica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Soberón, and provide a direct proof of the complex central transversal theorem. |
| title | Complex analogues of the Tverberg--Vrećica conjecture and central transversal theorems |
| topic | Combinatorics Algebraic Topology 52A35, 55N91, 55R91 |
| url | https://arxiv.org/abs/2408.14337 |