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Positive flow-spines and contact 3-manifolds

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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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Notes

  1. The positivity for flow-spines introduced in this paper is different from the one introduced in [19].

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, 223-8522, Japan

    Ippei Ishii

  2. Department of Mathematics, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kohoku, Yokohama, 223-8521, Japan

    Masaharu Ishikawa

  3. Department of Mathematics and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

    Yuya Koda

  4. Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Hironobu Naoe

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  1. Ippei Ishii
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  2. Masaharu Ishikawa
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Correspondence to Masaharu Ishikawa.

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Ishii, I., Ishikawa, M., Koda, Y. et al. Positive flow-spines and contact 3-manifolds. Annali di Matematica 202, 2091–2126 (2023). https://doi.org/10.1007/s10231-023-01314-1

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