Research on foliations, contact structures and Euler class

W. Thurston证明了一个没有Reeb分量的3流形上的叶形具有切线束的欧拉类满足一个不等式Thurston不等式的性质。另一方面，三球面上的Reeb叶形满足Thurston不等式，本研究之后的研究表明存在一类叶形，每个叶形都有Reeb分量并满足Thurston不等式。在2006年的研究中，我们得到了一类可旋叶理满足Thurston不等式的充分条件。此外，我们还揭示了瑟斯顿不等式不成立的一个方面。在关于接触结构对叶状的收敛性的研究中，我们研究了一个更精细的不等式，即Thurston不等式的相对版本，这一研究一直深入到2006年。事实上，对于可旋叶，我们证明了相对版本意味着绝对版本。对于接触结构，同样的说法是已知的，然而，它并不适用于叶状结构。同样在2007年，我们发现了用无穷阶欧拉类构造满足Thurston不等式的叶形的方法。在此之前，所有满足瑟斯顿不等式的叶都有平凡欧拉类。通过2006年的研究，我们确实找到了一个欧拉类为无限阶的可旋叶理。然后我们可以沿着Reeb分量进行Dehn手术，并在一定条件下对原可旋叶理进行Dehn手术，借助D. Gabai的缝合流形理论，我们可以得出在有限多例外情况下，结果满足无限阶欧拉类的Thurston不等式。

W. Thurston showed that a foliation on a 3-manifold which has no Reeb component enjoys the property that the Euler class of the tangent bundle satisfies an inequality, Thurston's inequality. On the other hand, the Reeb foliation on the three sphere satisfies Thurston's inequality and a previous research followed by this research showed that there is a class of foliations each of which has Reeb components and satisfi es Thurston's inequality.In the research in 2006, for a class of foliations which are called spinnable foliations, we obtained a sufficient condition for the foliation satisfying Thurston's inequality. Moreover, we revealed an aspect where Thurston's inequality does not hold. They are described by properties of the monodromy diffeomorphisms which determine the spinnable foliationsIn view of the research with respect to the convergence of contact structures to foliations, we studied a finer inequality, the relative version of Thurston's inequality, which deepens the research until 2006. In fact, for spinnable foliations we showed that the relative version implies the absolute version. The same statement for contact structures was known however, it does not hold in general for foliations. Also in 2007, we found the method to construct a foliation which satisfies Thurston's inequality with Euler class of infinite order. Until then, all foliations which satisfies Thurston's inequality have trivial Euler class. Indeed, we can find'a spinnable foliation whose Euler class is of infinite order by the research in 2006. Then we can perform Dehn surgery along the Reeb component and with certain condition on the original spinnable foliation we can conclude that with finitely many exceptions the resultant satisfies Thurston's inequality with Euler class of infinite order by virtue of D. Gabai's sutured manifold theory.