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Geometric criteria for overtwistedness


Authors: Roger Casals, Emmy Murphy and Francisco Presas
Journal: J. Amer. Math. Soc. 32 (2019), 563-604
MSC (2010): Primary 57R17; Secondary 53D10, 53D15
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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Additional Information

Roger Casals
Affiliation: Department of Mathematics, University of California Davis, Shields Avenue, Davis, California 95616
Email: casals@math.ucdavis.edu

Emmy Murphy
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: e_murphy@math.northwestern.edu

Francisco Presas
Affiliation: Instituto de Ciencias Matemáticas CSIC, C. Nicolás Cabrera, 13 28049 Madrid, Spain
Email: fpresas@icmat.es

DOI: https://doi.org/10.1090/jams/917
Received by editor(s): January 23, 2017
Received by editor(s) in revised form: September 12, 2018, and October 8, 2018
Published electronically: January 3, 2019
Additional Notes: The first author was supported by NSF grant DMS-1841913 and a BBVA Research Fellowship.
The second author was supported by NSF grant DMS-1510305 and a Sloan Research Fellowship.
The third author was supported by Spanish Research Projects SEV–2015–0554, MTM2016–79400–P, and MTM2015–72876–EXP
Article copyright: © Copyright 2019 American Mathematical Society