Reeb flows without simple global surfaces of section

TL;DR

This paper constructs Reeb flows on certain 3-spheres that lack simple global surfaces of section with fewer than a specified number of boundary components, demonstrating a new method for controlling the topology of these surfaces.

Contribution

It introduces a connected sum operation for open books to build Reeb flows with prescribed properties and proves stability under small perturbations.

Findings

Reeb flows without simple global surfaces of section with fewer than n boundary components.

Construction method using connected sums of open books.

Stability of the property under small perturbations.

Abstract

We construct, for any given positive integer $n$, Reeb flows on contact integral homology 3-spheres which do not admit global surfaces of section with fewer than $n$ boundary components. We use a connected sum operation for open books to construct such systems. We prove that this property is stable with respect to $C^{4+\epsilon}$-small perturbations of the Hamiltonian given on the symplectization.