On 2-convex non-orientable surfaces in four-dimensional Euclidean space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929649099997184 |
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| author | Bolotov, Dmitry V. |
| author_facet | Bolotov, Dmitry V. |
| contents | We prove that a 2-convex closed surface $S\subset E^4$ in the four-dimensional Euclidean space $E^4$, which is either $C^2$-smooth or polyhedral, provided that each vertex is incident to at most five edges, admits a mapping of degree one to a two-dimensional torus, where the degree is assumed to be $\mod 2$ if $S$ is nonorientable. As a corollary, we show that the projective plane and the Klein bottle do not admit such a 2-convex embedding in $E^4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19757 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On 2-convex non-orientable surfaces in four-dimensional Euclidean space Bolotov, Dmitry V. Geometric Topology We prove that a 2-convex closed surface $S\subset E^4$ in the four-dimensional Euclidean space $E^4$, which is either $C^2$-smooth or polyhedral, provided that each vertex is incident to at most five edges, admits a mapping of degree one to a two-dimensional torus, where the degree is assumed to be $\mod 2$ if $S$ is nonorientable. As a corollary, we show that the projective plane and the Klein bottle do not admit such a 2-convex embedding in $E^4$. |
| title | On 2-convex non-orientable surfaces in four-dimensional Euclidean space |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2412.19757 |