A new proof of Milnor-Wood inequality

Gaiane Panina; Timur Shamazov; Maksim Turevskii

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arXiv · 2412.14553

A new proof of Milnor-Wood inequality

Gaiane Panina·Timur Shamazov·Maksim Turevskii

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Abstract

The Milnor-Wood inequality states that if a (topological) oriented circle bundle over an orientable surface of genus $g$ has a smooth transverse foliation, then the Euler class of the bundle satisfies $$|\mathcal{E}|\leq 2g-2.$$ We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes $\mathcal{E}$ from the singularities of a quasisection. We also sketch two other proofs: one based on Poincar\`{e} rotation number theory, and the other of topological nature.

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BibTeX RIS

Gaiane Panina, Timur Shamazov, Maksim Turevskii. 2024-12-19. A new proof of Milnor-Wood inequality. https://arxiv.org/abs/2412.14553

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