Positive flow-spines and contact 3-manifolds

Ippei Ishii; Masaharu Ishikawa; Yuya Koda; Hironobu Naoe

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arXiv · 1912.05774

Positive flow-spines and contact 3-manifolds

Ippei Ishii·Masaharu Ishikawa·Yuya Koda·Hironobu Naoe

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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.

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Ippei Ishii, Masaharu Ishikawa, Yuya Koda, Hironobu Naoe. 2019-12-12. Positive flow-spines and contact 3-manifolds. https://arxiv.org/abs/1912.05774

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