Surveys in Differential Geometry

Volume 25 (2020)

Foliations, contact structures and their interactions in dimension three

Pages: 71 – 101



Vincent Colin (Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, Nantes, France)

Ko Honda (University of California, Los Angeles, Calif., U.S.A.)


We survey the interactions between foliations and contact structures in dimension three, with an emphasis on sutured manifolds and invariants of sutured contact manifolds. This paper contains two original results: the fact that a closed orientable irreducible $3$‑manifold $M$ with nonzero second homology carries a hypertight contact structure and the fact that an orientable, taut, balanced sutured $3$‑manifold is not a product if and only if it carries a contact structure with nontrivial cylindrical contact homology. The proof of the second statement uses the Handel–Miller theory of end-periodic diffeomorphisms of end-periodic surfaces.

2010 Mathematics Subject Classification

Primary 37B40, 57M50. Secondary 53C15.

V.C. was supported by ANR Quantact; K.H. was supported by NSF Grants DMS-1406564 and DMS-1549147.

Published 13 July 2022