Approximating C 0-foliations by contact structures

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 leaves of the foliation 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.