Geometry & Topology Seminar: "Smoothing Gabai’s Taut Foliations"

Abstract: Taut foliations of 3-manifolds have proven to be useful tools in studying their topology.  In the early to mid ’80’s, David Gabai systematically constructed taut, finite depth foliations on all compact 3-manifolds that could possibly admit such foliations and used them to answer many open and difficult questions in knot theory.  It has not been evident that these foliations are smooth.  A priori, they are only tangent to a continuous 2-plane field, but we have long conjectured that his methods can be used to produce C^1 foliations.  We have found a proof of this and have discovered  that,  under some mild additional conditions on the 3-manifold, the construction can be modified to produce C^\infty foliations.  This had not been conjectured and came as a bit of a surprise.

Host: Xiang Tang