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
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].
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.
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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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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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DOI: https://doi.org/10.1007/s12346-020-00368-3