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A Reeb Flow on the Three-Sphere Without a Disk-Like Global Surface of Section

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Abstract

We show that there are Reeb flows on the standard, tight three-sphere that do not admit global surfaces of section with one boundary component. In particular, the Reeb flows that we construct do not admit disk-like global surfaces of section. These Reeb flows are constructed using integrable systems, and a connected sum construction that extends the integrable system.

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Notes

  1. We allow the integrals to be dependent on a “small” subset to obtain more interesting topology. For example, there is always a “3-atom A” in Y in the sense of Bolsinov-Fomenko’s book [1, Section 3.5].

  2. A positive Dehn twist is often also called a right-handed Dehn twist. However, with the conventions in this paper, \(r d\varphi \) as Liouville form, the positive twist actually turns to the left. Note that the argument here only involves the invariant sets and not the direction of the twists. However, negative Dehn twists will yield an overtwisted three-sphere.

References

  1. Bolsinov, A., Fomenko, A.: Integrable Hamiltonian Systems: Geometry, Topology Classification. CRC Press, Boca Raton (2004). ISBN 0-415-29805-9

    Book  Google Scholar 

  2. Franks, J.: Geodesics on \(S^2\) and periodic points of annulus homeomorphisms. Invent. Math. 108(2), 403–418 (1992)

    Article  MathSciNet  Google Scholar 

  3. Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008). ISBN 978-0-521-86585-2

    Book  Google Scholar 

  4. Gompf, R., Stipsicz, A.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  5. Hofer, H., Wysocki, K., Zehnder, E.: The dynamics on three-dimensional strictly convex energy surfaces. Ann. Math. (2) 148(1), 197–289 (1998)

    Article  MathSciNet  Google Scholar 

  6. Hryniewicz, U.: Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere. J. Symplectic Geom. 12(4), 791–862 (2014)

    Article  MathSciNet  Google Scholar 

  7. Hryniewicz, U., Salomão, P.: On the existence of disk-like global sections for Reeb flows on the tight 3-sphere. Duke Math. J. 160(3), 415–465 (2011)

    Article  MathSciNet  Google Scholar 

  8. Kuperberg, K.: A smooth counterexample to the Seifert conjecture. Ann. Math. (2) 140(3), 723–732 (1994)

    Article  MathSciNet  Google Scholar 

  9. Lisi, S., Marinković, A., Niederkrüger, K.: Some properties of the Bourgeois contact structures, preprint arXiv:1801.00869

  10. Neto, O Manzoli, Massago, S., Saeki, O.: Open book structures on \((n-1)\)-connected \((2n+1)\)-manifolds. J. Math. Sci. Univ. Tokyo 13(4), 439–523 (2006)

    MathSciNet  MATH  Google Scholar 

  11. van Koert, O.: Lecture notes on stabilization of contact open books. Münster J. Math. 10(2), 425–455 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Weinstein, A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20(2), 241–251 (1991)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I thank Pedro Salomão for helpful comments. I was supported by NRF Grant NRF-2019R1A2C4070302, which was funded by the Korean Government.

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  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Building 27, Room 402 San 56-1, Sillim-dong, Gwanak-gu, Seoul, 08826, South Korea

    Otto van Koert

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  1. Otto van Koert
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Correspondence to Otto van Koert.

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van Koert, O. A Reeb Flow on the Three-Sphere Without a Disk-Like Global Surface of Section. Qual. Theory Dyn. Syst. 19, 36 (2020). https://doi.org/10.1007/s12346-020-00368-3

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