Quasisections of circle bundles and Euler class
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908309288648704 |
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| author | Panina, Gaiane Shamazov, Timur Turevskii, Maksim |
| author_facet | Panina, Gaiane Shamazov, Timur Turevskii, Maksim |
| contents | Let $ E \xrightarrow[\text{}]π B$ be an oriented circle bundle over an oriented closed surface $B$. A quasisection is a smooth surface ${Q}$ (either closed or bordered) mapped by a generic smooth mapping $q$ to $E$ such that $π\circ q({Q})=B$. In the paper we derive a local formula for the Euler number, that is, we show that Euler number (Euler class) of the bundle equals the sum of weights of (some of) singularities of a quasisection.We also prove the uniqueness of such a formula. The local formula is a close relative of M. Kazarian's formula which relates the Euler number and Morse bifurcations of a generic function defined on the total space $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasisections of circle bundles and Euler class Panina, Gaiane Shamazov, Timur Turevskii, Maksim Geometric Topology 57R22, 57R45 Let $ E \xrightarrow[\text{}]π B$ be an oriented circle bundle over an oriented closed surface $B$. A quasisection is a smooth surface ${Q}$ (either closed or bordered) mapped by a generic smooth mapping $q$ to $E$ such that $π\circ q({Q})=B$. In the paper we derive a local formula for the Euler number, that is, we show that Euler number (Euler class) of the bundle equals the sum of weights of (some of) singularities of a quasisection.We also prove the uniqueness of such a formula. The local formula is a close relative of M. Kazarian's formula which relates the Euler number and Morse bifurcations of a generic function defined on the total space $E$. |
| title | Quasisections of circle bundles and Euler class |
| topic | Geometric Topology 57R22, 57R45 |
| url | https://arxiv.org/abs/2410.22453 |