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2015 On representation of the Reeb graph as a sub-complex of manifold
Marek Kaluba, Wacław Marzantowicz, Nelson Silva
Topol. Methods Nonlinear Anal. 45(1): 287-307 (2015). DOI: 10.12775/TMNA.2015.015

Abstract

The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $\pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.

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Marek Kaluba. Wacław Marzantowicz. Nelson Silva. "On representation of the Reeb graph as a sub-complex of manifold." Topol. Methods Nonlinear Anal. 45 (1) 287 - 307, 2015. https://doi.org/10.12775/TMNA.2015.015

Information

Published: 2015
First available in Project Euclid: 30 March 2016

zbMATH: 1376.57037
MathSciNet: MR3365016
Digital Object Identifier: 10.12775/TMNA.2015.015

Rights: Copyright © 2015 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.45 • No. 1 • 2015
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