On representation of the Reeb graph as a sub-complex of manifold
DOI:
https://doi.org/10.12775/TMNA.2015.015Keywords
Reeb graph, critical point, gradient flowAbstract
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 $1$\nobreakdash-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 $M_g$.Downloads
Published
2015-03-01
How to Cite
1.
MARZANTOWICZ, Wacław, SILVA, Nelson and KALUBA, Marek. On representation of the Reeb graph as a sub-complex of manifold. Topological Methods in Nonlinear Analysis. Online. 1 March 2015. Vol. 45, no. 1, pp. 287 - 308. [Accessed 24 October 2024]. DOI 10.12775/TMNA.2015.015.
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