论文标题
歧管上平滑功能的Reeb空间
Reeb spaces of smooth functions on manifolds
论文作者
论文摘要
连续函数的REEB空间是级别集合的连接组件的空间。在本文中，我们首先证明，在有限的许多临界值的封闭歧管上，平滑函数的Reeb空间具有有限图的结构，而无需循环。我们还表明，没有循环的任意有限图可以实现为具有有限的许多临界值的封闭歧管上特定平滑函数的REEB空间，其中还可以预先签名相应的级别集。最后，我们表明，平滑的封闭连接的流形的连续图与有限连接的图形无环，可以诱导基本组之间的染色，并用自然商映射到具有有限的许多临界值的一定平滑函数的REEB空间，直到同型。
The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.
