The fully marked surface theorem

In his seminal 1976 paper, Bill Thurston observed that a closed leaf $S$ of a codimension‑$1$ foliation on a compact $3$‑manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on $[S]$, the homology class represented by $S$. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on $[S]$ equals up to sign the Euler characteristic of $S$ and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F}^\prime$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F}^\prime$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $F^\prime$ have the same Euler class.

In the same paper Thurston proved that taut foliations on closed hyperbolic 3‑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. This is the second of two papers that together give a negative answer to Thurston’s conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.

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Received 19 July 2018

Received revised 17 April 2020

Accepted 1 August 2020

Published 21 January 2021