Complex vs convex Morse functions and geodesic open books

TL;DR

This paper compares three different constructions of open books on the unit (co)tangent bundle of a surface, showing they are isotopic under certain conditions and analyzing their genus properties.

Contribution

It establishes isotopy equivalences among complex, contact, and dynamical open books on surface bundles based on Morse functions and divides.

Findings

Open books are pairwise isotopic if Morse functions are adapted.

None are planar if the surface has positive genus.

Identifies cases where open books have genus one pages.

Abstract

Suppose that $\Sigma$ 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 $\Sigma$, 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 $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.