Combinatorial modifications of Reeb graphs and the realization problem. (English) Zbl 1465.57099
Let \(M\) be a closed manifold, \(\dim M\ge2\), and \(G\) be a finite digraph with a so-called good orientation. The author considers the following realization problem: Is \(G\) homeomorphic to the Reeb graph \(R_f\) of some Morse function \(f\) on \(M\)? He proves that this is true if and only if \[b_1(G)\le\mathcal{R}(M),\] where \(b_1(G)\) is the cycle rank of the graph and \(\mathcal{R}(M)\) is the maximum cycle rank among all Reeb graphs of smooth functions on \(M\) with finite number of critical points. Moreover, any integer \(r\in[0,\mathcal{R}(M)]\) can be realized as the cycle rank of the Reeb graph of a Morse function on \(M\); in particular, the above inequality is exact. In addition, \(\mathcal{R}(M)=corank(\pi_1(M))\), the corank of the fundamental group of the manifold.
For surfaces, \(\dim M=2\), the author has proved a similar result even up to isomorphism in his previous work [Ł. P. Michalak, Topol. Methods Nonlinear Anal. 52, No. 2, 749–762 (2018; Zbl 1425.58022)].
For surfaces, \(\dim M=2\), the author has proved a similar result even up to isomorphism in his previous work [Ł. P. Michalak, Topol. Methods Nonlinear Anal. 52, No. 2, 749–762 (2018; Zbl 1425.58022)].
Reviewer: Irina Gelbukh (Ciudad de México)
MSC:
| 57M15 | Relations of low-dimensional topology with graph theory |
| 05C76 | Graph operations (line graphs, products, etc.) |
| 05C38 | Paths and cycles |
| 68U10 | Computing methodologies for image processing |
| 57R70 | Critical points and critical submanifolds in differential topology |
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