Research on foliations, contact structures and Euler class

W·瑟斯顿证明了3-流形上没有Reeb分支的叶层具有切丛的欧拉类满足一个不等式，即瑟斯顿不等式的性质。另一方面，三个球面上的Reeb叶层满足瑟斯顿不等式，而在此之前的研究表明，存在一类叶层，每个叶层都有Reeb分支且满足瑟斯顿不等式性.在2006年的研究中，对于一类称为可旋转叶层的叶层，我们得到了一个满足瑟斯顿不等式的叶层的充分条件.此外，我们揭示了瑟斯顿不平等不成立的一个方面。鉴于关于接触结构到叶层收敛的研究，我们研究了一个更精细的不等式，它是瑟斯顿不等式的相对版本，这一研究一直深化到2006年。事实上，对于可旋转的叶理，我们证明了相对版本意味着绝对版本。然而，对于接触结构来说，同样的说法是已知的，但对于叶理来说，它通常不成立。同样在2007年，我们发现了用无穷级欧拉类来构造满足瑟斯顿不等式的叶系的方法。在此之前，所有满足瑟斯顿不等式的叶都有平凡的欧拉类。事实上，通过2006年的研究，我们可以发现一种可旋转的叶理，它的欧拉级数是无穷大的。然后，我们可以沿着Reeb分量进行Dehn运算，在一定的条件下，我们可以得出结论，在有限个例外情况下，借助于D.Gabai的缝合流形理论，所得结果满足无限级Euler类的瑟斯顿不等式。

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.