Approximating $C^0$-foliations by contact structures

TL;DR

This paper demonstrates that co-orientable 2-dimensional foliations on closed orientable 3-manifolds can be approximated by contact structures, linking foliation properties to contact topology and Heegaard-Floer invariants.

Contribution

It establishes the approximation of $C^0$-foliations by contact structures and connects taut foliations to tight contact structures and Heegaard-Floer homology.

Findings

Any co-orientable 2-foliation can be approximated by contact structures.

Existence of taut $C^0$-foliation implies the presence of tight contact structures.

Manifolds with taut $C^0$-foliation are not $L$-spaces.

Abstract

We show that any co-orientable foliation of dimension two on a closed orientable $3$-manifold with continuous tangent plane field can be $C^0$-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 $C^0$-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 $C^0$-foliation of codimension-1 is not an $L$-space in the sense of Heegaard-Floer homology.