A new proof of Milnor-Wood inequality

TL;DR

The paper addresses the Milnor-Wood inequality for oriented circle bundles with transverse foliations over genus g surfaces. It presents a new proof based on a local formula that expresses the Euler class as a sum of weights of singular vertices of a quasisection, providing a concrete computational mechanism to bound E. It also sketches two alternative proofs via Poincaré rotation numbers and a purely topological approach, clarifying the landscape of methods for the bound |E| ≤ 2g−2. The work highlights a direct geometric interpretation of the Euler class through quasisection singularities and connects equality cases to dynamical foliations related to Anosov flows.

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 Poincarè rotation number theory, and the other of topological nature.

Figures (4)

• Figure 1: This is the preimage of a neighborhood of a singular vertex. We assume that the fibers are vertical. Here we have $k=0$, $n=1$, and $\mathcal{W}_{bb}(x)=1/6$. For its mirror image, $k=1$, $n=0$, and $\mathcal{W}_{bb}(x)=-1/6$
• Figure 2: The disks $D_1$ and $D_2$ are bounded by the bold lines.
• Figure 3: The surface $S_g=S_2$ is unfolded to an octagon. The projection of $\partial \mathcal{Q}$ onto $S_2$ is shown. The bold points indicate the singular vertices.
• Figure 4: The part of the quasisection $\mathcal{Q}_1$ lying in $\pi^{-1}(B)$ in assumption that each singular vertex contributes $1/6$.

Theorems & Definitions (2)

• Theorem 1
• Definition 1