全标记曲面定理
The fully marked surface theorem
摘要：
比尔·瑟斯顿在1976年的开创性论文中观察到，叶理的闭合叶S的欧拉示性数等于在[S]上求值的叶理的欧拉类，即由S表示的同调类。本文的主要结果是张紧叶理的一个逆命题：如果张紧叶理$\mathcal{F}$在[S]上的欧拉类等于S的欧拉示性数标的符号，且其下流形是双曲的，则存在另一张张紧叶理$\mathcal{F'}$，使得$S$与叶的并同调，且使得$\mathcal{F'}$的平面场与$\mathcal{F}$的平面场同伦.特别地，$\mathcal{F}$和$\mathcal{F'}$具有相同的欧拉类。在同一文件Thurston证明，拉紧foliations封闭双曲3-流形的欧拉类的范数至多为1，并证明，相反，任何积分上同调类的范数等于1是欧拉类的拉紧foliation。这是第二次的两个文件，共同给予否定的答案瑟斯顿猜想。在第一篇论文中，我们假设了本文的主要结果，构造了反例。
英文摘要：
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. 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 $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ 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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