发表机构
Beijing International Center for Mathematical Research（北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了闭图流形上所有切触叶状结构的欧拉类集合，并证明任意闭无球面图流形的闭对偶瑟斯顿单位球中的整系数类可虚拟实现为欧拉类，同时通过例子说明虚拟条件的必要性及实与整欧拉类为零的差异。
AI 中文摘要
在本文中，我们确定了闭图流形族上所有切触叶状结构的欧拉类集合。作为推论，对于任意闭的无球面图流形 $Y$，闭对偶瑟斯顿单位球中的每个整系数类都可以虚拟地实现为某个切触叶状结构的欧拉类。特别地，$Y$ 虚拟地承认具有消失欧拉类的切触叶状结构。我们给出了具有任意大第一贝蒂数的图流形例子，这些流形不承认具有消失实欧拉类的切触叶状结构。这表明虚拟条件是必要的。我们还提供了具有任意大第一贝蒂数的图流形例子，对于这些流形，每个切触叶状结构都有消失的实欧拉类，但没有任何一个具有消失的整欧拉类。这说明了实欧拉类为零与整欧拉类为零之间的细微差别。
英文摘要
In this paper, we determine the set of Euler classes of all taut foliations for a family of closed graph manifolds. As a consequence, for any closed aspherical graph manifold $Y$, every integral class in the closed dual Thurston unit ball can be virtually realized as the Euler class of some taut foliation. In particular, $Y$ virtually admits a taut foliation with vanishing Euler class. We present examples of graph manifolds with arbitrarily large first Betti number, which admit no taut foliations with vanishing real Euler classes. This shows that the virtual requirement is necessary. We also provide examples of graph manifolds with arbitrarily large first Betti number, for which every taut foliation has vanishing real Euler class, but none has vanishing integral Euler class. This illustrates the subtle difference between real Euler class zero and integral Euler class zero.
