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2018 Realization of a graph as the Reeb graph of a Morse function on a manifold
Łukasz Patryk Michalak
Topol. Methods Nonlinear Anal. 52(2): 749-762 (2018). DOI: 10.12775/TMNA.2018.029

Abstract

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.

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Łukasz Patryk Michalak. "Realization of a graph as the Reeb graph of a Morse function on a manifold." Topol. Methods Nonlinear Anal. 52 (2) 749 - 762, 2018. https://doi.org/10.12775/TMNA.2018.029

Information

Published: 2018
First available in Project Euclid: 9 August 2018

zbMATH: 07051691
MathSciNet: MR3915662
Digital Object Identifier: 10.12775/TMNA.2018.029

Rights: Copyright © 2018 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.52 • No. 2 • 2018
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