March 23, 2007

John Etnyre (Georgia Tech)

Title: The space of plane fields on a 3-manifold

Abstract: The space of plane fields on a 3-manifold contains several geometrically meaningful subspaces. Among these are the space of foliations, the space of contact structures and the space of "confoliations". In the mid 1990's Eliashberg and Thurston took the first step towards understanding how these subspaces sit in relation to one another in the space of all plane fields. Very roughly, they showed that any neighborhood of any point in the space of foliations contained a contact structure. This elegant result has had important implications in low dimensional topology. For example, it is part of Kronheimer and Mrowka's proof of the much studied conjecture that all nontrivial knots satisfy Property P. I will discuss recent work aimed at understanding these subspaces better. In particular I will comment on the following questions. Does the closure of every component of the space of contact structures intersect the space of foliations? Can foliations always be deformed into contact structures? How do properties of foliations (like being Reebless or taut) relate to properties of a contact structure (like being tight or overtwisted) near it?