A new proof of 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 $$|\mathcal{E}|\le 2g-2.$$ 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 Poincarè rotation number theory, and the other of topological nature.