Quasisections of circle bundles and Euler class

Gaiane Panina; Timur Shamazov; Maksim Turevskii

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arXiv · 2410.22453

Quasisections of circle bundles and Euler class

Gaiane Panina·Timur Shamazov·Maksim Turevskii

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Abstract

Let $ E \xrightarrow[\text{}]{\pi} 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 $\pi\circ q({Q})=B$. 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 $E$.

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BibTeX RIS

Gaiane Panina, Timur Shamazov, Maksim Turevskii. 2024-10-29. Quasisections of circle bundles and Euler class. https://arxiv.org/abs/2410.22453

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