Combinatorial modifications of Reeb graphs and the realization problem (Q2022614)

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 [\textit{Ł. P. Michalak}, Topol. Methods Nonlinear Anal. 52, No. 2, 749--762 (2018; Zbl 1425.58022)].