Approximating C 0-foliations by contact structures

2015/09/30 by Jonathan Bowden
Mathematics · #Codimension #Floer homology #Foliation (geology) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Plane (geometry) #Pure mathematics #Symplectic geometry #Tangent #Tangent space #math.DG #math.GT #math.SG

paper · pdf · doi:10.1007/s00039-016-0387-2

35 pages; 8 figures; Improved exposition following Referee's suggestions (To appear in Geom. Funct. Anal.)

arxiv created 2016/09/26 · arxiv updated 2016/09/27 · openalex publication_date 2016/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C⁰-approximated by both positive and negative contact structures unless all the leaves are simply connected. As applications we deduce that the existence of a taut C⁰-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C⁰-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.

Citations

• Tight contact structures on fibered hyperbolic 3-manifolds
• A Hölder continuous vector field tangent to many foliations
• The theory of foliations of codimension greater than one
• Holomorphic disks and genus bounds
• Contact perturbations of Reebless foliations are universally tight