C0 approximations of foliations

Kazez, William; Roberts, Rachel

doi:10.2140/gt.2017.21.3601

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$C^0$ approximations of foliations

William H Kazez and Rachel Roberts

Geometry & Topology 21 (2017) 3601–3657

DOI: 10.2140/gt.2017.21.3601

Abstract

Suppose that $ℱ$ is a transversely oriented, codimension-one foliation of a connected, closed, oriented $3$ –manifold. Suppose also that $ℱ$ has continuous tangent plane field and is taut; that is, closed smooth transversals to $ℱ$ pass through every point of $M$ . We show that if $ℱ$ is not the product foliation $S^{1} \times S^{2}$ , then $ℱ$ 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.

Keywords

taut foliation, holonomy, contact topology, weakly symplectically fillable, universally tight

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 53D10

References

Publication

Received: 3 January 2016

Accepted: 30 January 2017

Published: 31 August 2017

Proposed: Cameron Gordon

Seconded: Yasha Eliashberg, András I Stipsicz

Authors

William H Kazez
Department of Mathematics University of Georgia Athens, GA United States

Rachel Roberts
Department of Mathematics Washington University St Louis, MO United States

Volume 21, issue 6 (2017)