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.
Similar content being viewed by others
Notes
The positivity for flow-spines introduced in this paper is different from the one introduced in [19].
References
Benedetti, R., Petronio, C.: Branched Standard Spines of 3-Manifolds. Lecture Notes in Mathematics, vol. 1653. Springer, Berlin (1997)
Benedetti, R., Petronio, C.: Branched spines and contact structures on 3-manifolds. Ann. Mat. Pure Appl. 178(4), 81–102 (2000)
Bennequin, D.: Entrelacements et équations de Pfaff. Astérique 107–108, 87–161 (1983)
Colin, V., Giroux, E., Honda, K.: Finitude homotopique et isotopique des structures de contact tendues. Publ. Math. Inst. Hautes Études Sci. 109, 245–293 (2009)
Eliashberg, Y.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98(3), 623–637 (1989)
Eliashberg, Y.: Filling by Holomorphic Discs and Its Applications, Geometry of Low-Dimensional Manifolds, 2 (Durham, 1989). London Mathematical Society Lecture Note Series, vol. 151, pp. 45–67. Cambridge University Press, Cambridge (1990)
Eliashberg, Y.: Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier Grenoble 42, 165–192 (1992)
Eliashberg, Y., Thurston, W.: Confoliations, University Lecture Series 13. American Mathematical Society, Providence (1998)
Endoh, M., Ishii, I.: A new complexity for 3-manifolds. Jpn. J. Math. N.S. 31, 131–156 (2005)
Etnyre, J.B.: Lectures on Open Book Decompositions and Contact Structures, Floer Homology, Gauge Theory, and Low-Dimensional Topology, pp. 103–141. Clay Mathematical Proceedings, vol. 5. American Mathematical Society, Providence (2006)
Geiges, H.: An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)
Giroux, E.: Convexité en topologie de contact. Comment. Math. Helvetici 66, 637–677 (1991)
Giroux, E.: Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 405–414. Higher Ed. Press, Beijing (2002)
Gray, J.W.: Some global properties of contact structures. Ann. Math. 2(69), 421–450 (1959)
Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)
Ikeda, H.: Acyclic fake surfaces. Topology 10, 9–36 (1971)
Ikeda, H.: Acyclic fake surfaces which are spines of 3-manifolds. Osaka Math. J. 9, 391–408 (1972)
Ikeda, H., Inoue, Y.: Invitation to DS-diagram. Kobe J. Math. 2, 169–186 (1985)
Ishii, I.: Flows and spines. Tokyo J. Math. 9, 505–525 (1986)
Ishii, I.: Moves for flow-spines and topological invariants of \(3\)-manifolds. Tokyo J. Math. 15, 297–312 (1992)
Koda, Y.: Branched spines and Heegaard genus of 3-manifolds. Manuscr. Math. 123(3), 285–299 (2007)
Lutz, R.: Structures de contact sur les fibre’s principaux en cercles de dimension 3. Ann. Inst. Fourier 27–3, 1–15 (1977)
Martinet, J.: Formes de contact sur les variétiés de dimension 3. Lect. Notes Math. 209, 142–163 (1971)
Matveev, S.: Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics, vol. 9. Springer, Berlin (2003)
Ozbagci, B., Stipsicz, A.I.: Surgery on Contact 3-Manifolds and Stein Surfaces. Bolyai Society Mathematical Studies, vol. 13. Springer (2004)
Petronio, C.: Generic flows on \(3\)-manifolds. Kyoto J. Math. 55(1), 143–167 (2015)
Thurston, W., Winkelnkemper, H.: On the existence of contact forms. Proc. Am. Math. Soc. 52, 345–347 (1975)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-023-01314-1