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.
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Bowden, J. Approximating C 0-foliations by contact structures. Geom. Funct. Anal. 26, 1255–1296 (2016). https://doi.org/10.1007/s00039-016-0387-2
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DOI: https://doi.org/10.1007/s00039-016-0387-2