$C^0$ Approximations of foliations

Abstract

Suppose that $\mathcal{F}$ is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that $\mathcal{F}$ has continuous tangent plane field and is {\sl taut}; that is, closed smooth transversals to $\mathcal{F}$ pass through every point of $M$. We show that if $\mathcal{F}$ is not the product foliation $S^1\times S^2$, then $\mathcal{F}$ can be $C^0$ approximated by weakly symplectically fillable, universally tight, contact structures. This extends work of Eliashberg-Thurston on approximations of taut, transversely oriented $C^2$ foliations to the class of foliations that often arise in branched surface constructions of foliations. This allows applications of contact topology and Floer theory beyond the category of $C^2$ foliated spaces.

arXiv comment56 pages, 6 figures. Definitions have been added and clarified, and citations have been updated

References in corpus (6)

• Holomorphic disks and genus bounds
• Laminar Branched Surfaces in 3-manifolds
• Approximating $C^0$-foliations by contact structures
• Leafwise smoothing laminations
• Taut foliations in knot complements
• Taut foliations in surface bundles with multiple boundary components

Cited by in corpus (15)

• Floer Simple Manifolds and L-Space Intervals
• L-spaces, taut foliations, and graph manifolds
• Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds
• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• Taut Foliations, Positive 3-Braids, and the L-Space Conjecture
• Thin links and Conway spheres
• L-spaces, taut foliations and the Whitehead link
• $C^{1,0}$ Foliation Theory
• Reeb flows transverse to foliations
• Taut foliations, braid positivity, and unknot detection
• Taut foliations leafwise branch cover S^2
• Taut foliations from knot diagrams
• L-spaces, taut foliations and fibered hyperbolic two-bridge links
• Chainmail links and L-spaces
• Small Heegaard genus and SU(2)