Positive flow-spines and contact 3-manifolds

TL;DR

This paper introduces positive flow-spines in 3-manifolds and proves they support unique contact structures, extending the understanding of the relationship between flow-spines and contact topology.

Contribution

It defines positivity for flow-spines and establishes that positive flow-spines support a unique contact structure, a novel connection in 3-manifold topology.

Findings

Positive flow-spines support a unique contact structure.

Positivity condition is essential for the existence of the supported contact structure.

Extension of Thurston-Winkelnkemper theorem to flow-spines.

Abstract

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.