Realization of a graph as the Reeb graph of a Morse function on a manifold. (English) Zbl 1425.58022
The author studies the problem of realization of a given finite graph as the Reeb graph \(\mathcal{R}(f)\) of a smooth function \(f : M \rightarrow \mathbb{R}\) with finitely many critical points, where \(M\) is a closed manifold. He proves the following interesting result: For every \(n\geq 2\) and for every graph \(\Gamma\) admitting the so-called good orientation there exist an \(n\)-manifold \(M\) and a Morse function \(f : M \rightarrow \mathbb{R}\) such that its Reeb graph \(\mathcal{R}(f)\) is isomorphic to \(\Gamma\) (Theorem 4.3).
This property extends the previous results of V. V. Sharko [Methods Funct. Anal. Topol. 12, No. 4, 389–396 (2006; Zbl 1114.57034)] and Y. Masumoto and O. Saeki [Kyushu J. Math. 65, No. 1, 75–84 (2011; Zbl 1277.58022)].
The paper is organized into five sections as follow: Introduction, Basic properties of Reeb graphs, Number of cycles in Reeb graphs, Realization by a Morse function, Solution of the main problem for surfaces.
This property extends the previous results of V. V. Sharko [Methods Funct. Anal. Topol. 12, No. 4, 389–396 (2006; Zbl 1114.57034)] and Y. Masumoto and O. Saeki [Kyushu J. Math. 65, No. 1, 75–84 (2011; Zbl 1277.58022)].
The paper is organized into five sections as follow: Introduction, Basic properties of Reeb graphs, Number of cycles in Reeb graphs, Realization by a Morse function, Solution of the main problem for surfaces.
Reviewer: Dorin Andrica (Riyadh)
MSC:
| 58K05 | Critical points of functions and mappings on manifolds |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 58K65 | Topological invariants on manifolds |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 57R70 | Critical points and critical submanifolds in differential topology |
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