Intersection norms on surfaces and Birkhoff cross sections

TL;DR

This paper introduces a new (semi-)norm on the first homology of surfaces based on collections of curves, and classifies Birkhoff cross sections for geodesic flows on constant curvature surfaces using these norms.

Contribution

It defines a novel (semi-)norm related to curve collections and provides a classification of Birkhoff cross sections via integer points in the dual norm's interior.

Findings

Dual norm unit ball's convex hull of integer points

Integer points classify isotopy classes of Birkhoff cross sections

Birkhoff cross sections produce open-book decompositions

Abstract

For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these points in terms of certain coorientations of the original collection of curves. Our main result is a classification statement: when the surface has constant curvature and the curves are geodesics, integer points in the interior of the dual unit ball classify isotopy classes of Birkhoff cross sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the collection of geodesics. Birkhoff cross sections in particular yield open-book decompositions of the unit tangent bundle.