The fully marked surface theorem

Abstract:

In his seminal 1976 paper Bill Thurston observed that a closed leaf \(S\) of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on \([S]\), the homology class represented by \(S\). We give a converse for taut foliations: if the underlying manifold is hyperbolic and if the Euler class of a taut foliation \(F\) evaluated on \([S]\) equals, up to sign, the Euler characteristic of \(S\), then there exists another taut foliation \(F'\) such that \(S\) is homologous to a union of leaves and such that the plane field of \(F'\) is homotopic to that of \(F\). In particular, \(F\) and \(F'\) have the same Euler class.

In the same paper Thurston proved that taut foliations on closed hyperbolic three-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. Work of Yazdi, together with our main result, give a negative answer to Thurston's conjecture.

This is joint work with Mehdi Yazdi.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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