Intersection norms on surfaces and Birkhoff cross sections

TL;DR

This work introduces intersection norms on the first homology of a surface, encoded by a multicurve $\gamma$, and connects their dual unit balls to Eulerian coorientations via eikonal-function methods. For hyperbolic surfaces, integer interior points of the dual ball correspond to isotopy classes of negative-boundary Birkhoff cross sections for the geodesic flow on the unit tangent bundle, realized concretely by Birkhoff–Brunella surfaces $S^{BB}(\eta)$. The results situate these cross sections within the Schwartzman–Fried–Sullivan framework and link them to Thurston’s norm theory, extending to orientable 2-orbifolds and providing a constructive appendix on Thurston’s theorem for integral seminorms. The work thus yields an explicit, combinatorial classification of geodesic-flow Birkhoff sections with symmetric boundary, and opens avenues for generalizations to broader flows and orbifold contexts. Overall, the paper blends discrete intersection theory, dynamical systems, and 3-manifold topology to give a sharp, computable dictionary between curve data on surfaces and global cross sections in associated flows.

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.

Figures (10)

• Figure 1: Illustration of Theorems \ref{['T:Classification']} and \ref{['T:Coor']} in the case of $\Sigma$ a torus (with an abuse since Theorem \ref{['T:Classification']} deals with higher-genus surfaces, whose homology has dimension $\ge 4$). On the left, a multicurve $\gamma$ on $\Sigma$ consisting of four geodesics, and an Eulerian coorientation (blue arrows). Seen as a graph, $\gamma$ has 5 vertices and 10 edges. On the right, the dual unit ball $B^*_{{\bf x}_\gamma}$ of the associated intersection norm. The empty circle denotes the origin. The big dots denote those classes in $\mathrm{H}^1(\Sigma; \mathbb{Z})$ congruent to $[\gamma]_2$ mod 2. Among these classes, 10 (in blue, green and red) are in the dual unit ball $B^*_{{\bf x}_\gamma}$ and correspond to all cohomology classes of Eulerian coorientations of $\gamma$ (Theorem \ref{['T:Coor']}). For example, the class corresponding to the blue coorientation is the blue point. The blue and green points lie in the interior of $B^*_{{\bf x}_\gamma}$, hence describe two isotopy classes of Birkhoff cross sections for the geodesic flow bounded by $-{\overset\leftrightarrow{\gamma}}$. If the genus of $\Sigma$ was at least 2, there would be no other isotopy class of Birkhoff cross section for the geodesic flow (Theorem \ref{['T:Classification']}). The 8 red points are on the boundary of $B^*_{{\bf x}_\gamma}$ and correspond to classes of surfaces transverse to the geodesic flow, but not intersecting every orbit, and bounded by $-{\overset\leftrightarrow{\gamma}}$.
• Figure 2: A genus 3 surface with a multicurve $\gamma$ made of four closed curves (black). On the left the curve $\alpha_1$ (orange and bold) is transverse to $\gamma$ and intersects it three times. On the right $\alpha_2$ (red) is homologous to $\alpha_1$ since their difference bounds a subsurface, namely the right hemisurface. The curve $\alpha_2$ intersects $\gamma$ only once. This number cannot be reduced to $0$ in the same homology class, hence $\alpha_2$ is ${\bf x}_\gamma$-realizing and we have ${\bf x}_\gamma([\alpha_1])={\bf x}_\gamma([\alpha_2])=|\{\alpha_2^{-1}(\gamma)\}|=1.$
• Figure 3: A torus with a collection $\gamma$ (black) made of four curves, two vertical and two horizontal. The curve $\alpha$ (red and bold) intersects $\gamma$ in 10 points. It is the smallest number for a curve whose homology class is $(4,1)$, so that ${\bf x}_\gamma(4,1)=10$. The norm ${\bf x}_\gamma$ is actually given by ${\bf x}_\gamma((p,q))=2|p|+2|q|$ in the canonical coordinates.
• Figure 4: A piece of a multicurve $\gamma$ (black). A coorientation $\eta$ of $\gamma$ is indicated with blue arrows. A path $\alpha$ transverse to $\gamma$ is shown (purple and dotted). The pairing $\langle \eta ,\alpha \rangle$ equals $-1+2=+1$ on this example.
• Figure 5: A part of the multicurve $\widetilde{\gamma}$ (black and thin). Assume that the set $D$ consists of three points $y_1, y_2, y_3$ (red, green and blue dots) with prescribed values $f(y_1)=0, f(y_2)=2$ and $f(y_3)=-1$. Considering a fourth point $x$ (purple), we see that we have $I_{x,y_1}=[-1,5]$, $I_{x,y_2}=[-1,1]$ and $I_{x,y_3}=[-3,1]$. In particular these three intervals intersect, and one can set $\overline f(x) = 1$.

Theorems & Definitions (92)

• Theorem A
• Proposition B
• Remark 1
• Theorem C
• Proposition D
• Remark 2
• Remark 3
• Remark 4
• Theorem E
• Definition 1.1