A new proof of Milnor-Wood inequality

2024/12/19 by Gaiane Panina, Panina, Gaiane, Timur Shamazov +3
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2412.14553

openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 |E|≤ 2g-2. We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes 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.

Related

• Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes
• Quasisections of circle bundles and Euler class
• The generalized Chern conjecture for manifolds that are locally a product of surfaces
• Affine transverse foliations in sphere bundles
• The hyperbolic meaning of the Milnor-Wood inequality