Positive flow-spines and contact 3-manifolds

Ippei Ishii, Masaharu Ishikawa, Yuya Koda, Hironobu Naoe

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)2091-2126
Number of pages36
JournalAnnali di Matematica Pura ed Applicata
Issue number5
Publication statusPublished - 2023 Oct


  • 3-dimensional manifold
  • Contact structure
  • Flow
  • Polyhedron
  • Spine

ASJC Scopus subject areas

  • Applied Mathematics


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