Geometric criteria for overtwistedness

Open access version at https://arxiv.org/abs/1503.06221In 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.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