A Borsuk--Ulam theorem for well separated maps
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916080921870336 |
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| author | Crabb, M. C. |
| author_facet | Crabb, M. C. |
| contents | Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam theorem asserts that the image of the map $(f_1,\ldots ,f_m) :S(V)\to R^m$ contains $0=(0,\ldots ,0)$. Pursuing ideas in papers of Bárány, Hubard and Jéronimo (2008) and Frick and Wellner (2023), we show that a certain separation property will guarantee that the image contains an $m$-cube. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Borsuk--Ulam theorem for well separated maps Crabb, M. C. Algebraic Topology 52A20, 52A38, 55M20, 55M25, 57S17 Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam theorem asserts that the image of the map $(f_1,\ldots ,f_m) :S(V)\to R^m$ contains $0=(0,\ldots ,0)$. Pursuing ideas in papers of Bárány, Hubard and Jéronimo (2008) and Frick and Wellner (2023), we show that a certain separation property will guarantee that the image contains an $m$-cube. |
| title | A Borsuk--Ulam theorem for well separated maps |
| topic | Algebraic Topology 52A20, 52A38, 55M20, 55M25, 57S17 |
| url | https://arxiv.org/abs/2401.02209 |