Realization of a graph as the Reeb graph of a Morse function on a manifold
| Autor: | Michalak, Łukasz Patryk |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749-762 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.12775/TMNA.2018.029 |
| Popis: | We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces. Comment: Major revision, to appear in Topol. Methods Nonlinear Anal |
| Databáze: | arXiv |
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