论文标题
完全标记的表面定理
The fully marked surface theorem
论文作者
论文摘要
瑟斯顿（Thurston）在1976年的1976年论文法案中观察到，叶子的闭叶s的特征是在[s]上评估的叶子类等于签名的欧拉类别，S。由S代表的同源性类别。 S和基础歧管是双曲线的，然后存在另一个拉紧的叶子$ \ MATHCAL {f'} $，使得$ s $与叶子结合是同源的，因此$ \ nathcal {f'} $的平面场与$ \ mathcal {f} $的$ \ nathcal {f'} $是同型。特别是，$ \ Mathcal {f} $和$ \ Mathcal {f'} $具有相同的Euler类。 在同一篇论文中，瑟斯顿证明，封闭双曲线3个序列上的绷紧叶子最多具有欧拉类别的规范类，并猜想，相反，任何具有等于norm等等的积分共同体类别都是绷紧叶的欧拉类。这是两篇论文中的第二篇，共同对瑟斯顿的猜想产生了负面答案。在第一篇论文中，假设本文的主要结果构建了反例。
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.
