A new proof of Milnor-Wood inequality

TL;DR

This paper presents a new proof of the Milnor-Wood inequality for circle bundles over surfaces, utilizing a local formula involving singularities of quasisections, and sketches alternative proofs based on rotation number theory and topology.

Contribution

The authors introduce a novel proof of the Milnor-Wood inequality using a local formula for the Euler class derived from quasisection singularities, complementing existing proofs.

Findings

New proof based on local formula and quasisection singularities

Alternative proofs using Poincaré rotation number theory and topological methods

Reinforces the validity of the Milnor-Wood inequality

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.