$C^0$ 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 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 and 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.