$C^0$ Approximations of foliations

TL;DR

This paper proves that certain taut, transversely oriented, codimension one foliations on 3-manifolds can be approximated by contact structures with specific tightness and fillability properties, extending previous smooth results to continuous cases.

Contribution

It extends Eliashberg-Thurston's approximation results from $C^2$ foliations to those with continuous tangent plane fields, broadening applications in contact topology.

Findings

Foliations not equal to $S^1\times S^2$ can be approximated by tight, fillable contact structures.

This approximation applies to foliations arising from branched surface constructions.

Enables new applications of contact topology and Floer theory to less smooth 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.