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.

中文翻译：

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完全标记的曲面定理

比尔·瑟斯顿（Bill Thurston）在1976年的开创性论文中观察到，紧紧的$ 3 $流形上的余叶$ 1 $的余量叶子$ S $的欧拉特性等于对$ [ S] $，以$ S $表示的同源性类。本文的主要结果是张紧叶面的反面：如果对$ [S] $评估的张紧叶面的Euler类$ \ mathcal {F} $等于标志着$ S $的Euler特征和基础流形是双曲线，然后存在另一个拉紧叶$ \ mathcal {F} ^ \ prime $，使得$ S $与叶子并集同源，并且$ \ mathcal {F} ^ \ prime $的平面场与$ \ mathcal {F } $。特别地，$ \ mathcal {F} $和$ F ^ \ prime $具有相同的Euler类。在同一篇论文中，Thurston证明了封闭双曲3流形上的紧拉叶理最多具有Euler范数范数，并且推测相反，任何范数等于1的积分同调类都是紧拉叶理的Euler类。这是两篇论文的第二篇，对瑟斯顿的猜想给出了否定的答案。在第一篇论文中，假设本文的主要结果构成了反例。