Geometric criteria for overtwistedness

Casals, Roger; Murphy, Emmy; Presas, Francisco

doi:10.1090/jams/917

Geometric criteria for overtwistedness
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by Roger Casals, Emmy Murphy and Francisco Presas

J. Amer. Math. Soc. 32 (2019), 563-604

DOI: https://doi.org/10.1090/jams/917

Published electronically: January 3, 2019

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Abstract:

In this article we establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then we characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, we provide several applications of these geometric criteria.

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