Complex vs convex Morse functions and geodesic open books

TL;DR

The paper unifies three seemingly distinct constructions of open books on $ST\Sigma$, $ST^*\Sigma$, and $V(\Sigma)$—A'Campo-Ishikawa, Giroux, and geodesic/cogeodesic open books—by showing that, for an admissible divide $P$ and an ordered Morse function $f$ adapted to $P$, the resulting open books are pairwise isotopic under natural identifications. It develops the complexification of $f$ to produce achiral Lefschetz fibrations, leverages Giroux’s convexity framework to relate contact and open-book data, and uses Birkhoff cross sections to connect the dynamical/geodesic viewpoint with the topological open-book framework. A key technical contribution is a simple isotopy criterion: if two open books share an isotopic page, they are isotopic, allowing a four-page coincidence that ties the A'Campo-Ishikawa, Giroux, and geodesic constructions together. The genus-one and minimal-genus phenomena are analyzed in detail, revealing explicit divisions that yield genus-one Lefschetz fibrations with precise binding counts and monodromy descriptions, thereby clarifying the landscape of open books on $ST^*\Sigma$ and their contact-geometric implications.

Abstract

Suppose that $Σ$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $Σ$, having complex, contact, and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $Σ$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $Σ$. Moreover, we observe that if $Σ$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

Figures (10)

• Figure 1: An admissible divide $P$ (in red) on a genus 2 surface, a black-and-white coloring of the complement $\Sigma\setminus P$, and some level sets of an ordered Morse function adapted to $P$. The divide $P$ consists of six curves and includes seven double points.
• Figure 2: Some of the pages $S_\theta$ of the A'Campo-Ishikawa open book for $ST\Sigma$, viewed as sets of vectors tangent to $\Sigma$.
• Figure 3: The pages $S_\theta$ of the A'Campo-Ishikawa open book for $ST\Sigma$ around the fiber of a double point of a divide $P$, for $\theta=\frac{k\pi}{4}$ with $k=0, \dots, 7$. The arrows correspond to the generators of the geodesic flow on $ST\Sigma$, in the case where the divide $P$ is geodesic. They are tangent to the link $L(P)$ and transverse to the interiors of all pages.
• Figure 4: A decomposition of the page $S_{\frac{\pi}{2}}$ in the A'Campo-Ishikawa's open book into rectangles, each of which corresponds to one edge of the divide. The figure shows the four rectangles adjacent to the fiber of a double point of the divide. Each rectangle consists of one face, 6 edges, and 6 vertices. The horizontal edges (2 per rectangle) correspond to the link of the divide and form the boundary of $S_{\frac{\pi}{2}}$. The vertical edges (4 per rectangle) are glued 2 by 2. The vertices are glued 3 by 3. Hence the contribution of each rectangle to the Euler characteristics of $S_{\frac{\pi}{2}}$ is $1-2-4\cdot\frac{1}{2}+6\cdot\frac{1}{3}=-1$.
• Figure 5: Flow lines of a gradient-like vector field for an ordered Morse function associated to a divide on a surface.

Theorems & Definitions (37)

• Theorem 1.1
• Proposition 1.2
• Remark 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 2.1
• Theorem 2.2: A'Campo-Ishikawa