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.

中文翻译：

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流形上光滑函数的Reeb空间

连续功能的Reeb空间是级别集的连接组件的空间。在本文中，我们首先证明具有有限多个临界值的闭合流形上的光滑函数的Reeb空间具有不带环的有限图的结构。我们还表明，可以将任意有限的无环图实现为具有有限多个临界值的闭合流形上某个平滑函数的Reeb空间，其中相应的水平集也可以预先指定。最后，我们表明，利用自然商图可以识别出具有一定临界值的某个光滑函数的Reeb空间的自然商图，该光滑图闭合连通的流形与一个无环的有限连通图的连续映射，该环不会引起基本基团之间的同质性。 ，直到同伦。