Surfaces of section for Seifert fibrations

TL;DR

We classify global surfaces of section for flows on closed 3‑manifolds that define Seifert fibrations and establish a comprehensive framework for d‑sections, including a necessary divisibility criterion, descent to Z_d‑quotients, and a Riemann–Hurwitz based genus computation. The approach links the dynamics to a branched covering picture over the Hopf fibration and to algebraic curves in weighted projective planes, enabling explicit genus and boundary predictions through weighted homogeneous polynomials and degree–genus formulas. The results yield explicit classifications for Seifert fibrations of S^3, reveal a tight correspondence between d‑sections and weighted algebraic curves, and show that every Seifert manifold admits a d‑section for some d, with positive d‑sections existing precisely when the Euler number is non‑positive after choosing an orientation. Together these findings provide a geometric bridge between 3‑manifold dynamics, Seifert topology, and weighted projective algebraic geometry, with practical implications for Reeb dynamics and Besse flows.

Abstract

We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings -- one way or the other -- between surfaces of section for the Hopf flow and those for any other Seifert fibration of the 3-sphere, and we relate these surfaces of section to algebraic curves in weighted complex projective planes.

Theorems & Definitions (30)

• Lemma 4.1
• proof
• Theorem 4.2
• Example 4.3
• proof : Proof of Theorem \ref{['thm:1-section']}
• Lemma 5.1
• Remark 5.2
• proof : Proof of Lemma \ref{['lem:d-section']}
• Proposition 5.3
• Theorem 5.4