The fully marked surface theorem (Q2663758)

In an influential paper published in 1976 (revised version published in 1986, [\textit{W. P. Thurston}, Mem. Am. Math. Soc. 339, 99--130 (1986; Zbl 0585.57006)]), William Thurston observed that a closed leaf \(S\) of a codimension-one 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 of this result for taut foliations. The authors prove that 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 \(\mathcal{F}^\prime\) have the same Euler class. In the same paper cited above, 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. In this paper, the authors give a negative answer to Thurston's conjecture.