In 1976, Thurston proved that taut foliations on closed hyperbolic $3$-manifolds have Euler class of norm at most $1$, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counter-examples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

中文翻译：

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关于瑟斯顿的欧拉一类猜想

1976年，瑟斯顿（Thurston）证明了封闭双曲线$ 3 $流形上的紧拉叶理具有至多1美元的Euler范数范本，并推测相反地，范数等于1的任何第二同调整阶都是紧拉叶理的Euler类。这是两篇论文系列中的第一篇，它们共同为瑟斯顿的猜想提供了否定的答案。在此，反例是根据完全标记的曲面定理构造的。在第二篇论文中，与David Gabai联合，给出了完全标记表面定理的证明。