We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number \(\mathcal {R}(M)\), i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group \(\pi _1(M)\), thus extending a previous result of Gelbukh to the non-orientable case.

中文翻译：

---

Reeb图的组合修改及实现问题

我们证明，直到同胚为止，任何在方向和循环秩上受自然必要条件约束的图都可以实现为给定闭合流形M上的摩尔斯函数的Reeb图 。一路上，我们表明，力波数 \（\ mathcal {R}（M）\） ，即，功能上的所有力波的曲线图中的最大周期秩中号，等于基本组的corank  \（\ PI _1（M）\），从而将Gelbukh的先前结果扩展到不可定向的情况。