Foliations, contact structures and their interactions in dimension three

2018/11/22 by Vincent Colin, Colin, Vincent, Ko Honda +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1811.09148

openalex publication_date 2018/11/22 · openalex created_date 2018/11/29 · openalex updated_date 2026/07/28

Abstract

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 homol-ogy 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.

Citations

• End periodic surface homeomorphisms and 3-manifolds
• Endperiodic automorphisms of surfaces and foliations
• On L-spaces and left-orderable fundamental groups
• The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus
• The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II

Related

• Contact structures on 3-manifolds are deformations of foliations
• Existence and classification of b-contact structures
• The topology and geometry of contact structures in dimension three
• Bordered Floer homology and contact structures
• Existence and classification of overtwisted contact structures in all dimensions