[1907.06025] Convex hypersurface theory in contact topology Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Symplectic Geometry arXiv:1907.06025 (math) [Submitted on 13 Jul 2019 ( v1 ), last revised 28 Apr 2026 (this version, v4)] Title: Convex hypersurface theory in contact topology Authors: Joseph Breen , Austin Christian , Ko Honda , Yang Huang View a PDF of the paper titled Convex hypersurface theory in contact topology, by Joseph Breen and Austin Christian and Ko Honda and Yang Huang View PDF HTML (experimental) Abstract: We lay the foundations of convex hypersurface theory in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also prove that a $C^0$-generic family of mutually disjoint closed hypersurfaces parametrized by $t\in[0,1]$ is convex except at finitely many times $t_1,\dots,t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As an application, we prove the existence of compatible (relative) open book decompositions for contact manifolds. Comments: V4: Added two coauthors: Joseph Breen and Austin Christian; the part on contact submanifolds has been removed and will be written more carefully in a separate paper; the proof of the bypass-bifurcation correspondence has been expanded and the paper is now essentially self-contained with a 24-page appendix on bypasses in higher dimensions Subjects: Symplectic Geometry (math.SG) Cite as: arXiv:1907.06025 [math.SG] (or arXiv:1907.06025v4 [math.SG] for this version) https://doi.org/10.48550/arXiv.1907.06025 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ko Honda [ view email ] [v1] Sat, 13 Jul 2019 08:06:14 UTC (850 KB) [v2] Tue, 23 Jul 2019 17:25:03 UTC (850 KB) [v3] Sun, 25 Jun 2023 06:49:00 UTC (551 KB) [v4] Tue, 28 Apr 2026 21:01:37 UTC (1,454 KB) Full-text links: Access Paper: View a PDF of the paper titled Convex hypersurface theory in contact topology, by Joseph Breen and Austin Christian and Ko Honda and Yang Huang View PDF HTML (experimental) TeX Source view license Current browse context: math.SG < prev | next > new | recent | 2019-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from