Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups (Q6614521)

The \textit{Reeb graph} \(\mathcal{R}(f)\) of a function \(f:M\to\mathbf{R}\) is a topological space obtained by contracting connected components of its level sets to points, endowed with the quotient topology. For a Morse function on a compact manifold, \(\mathcal{R}(f)\) is a finite graph, so its fundamental group is a free group, \(\pi_1(\mathcal{R}(f))=F_r\). The quotient map \(q_f:M\to\mathcal{R}(f)\) induces the epimorphism of the fundamental groups \(q_{f\sharp}:\pi_1(M)\to F_r\), called the \textit{Reeb epimorphism} of \(f\).\N\NThe authors prove that conversely, it is possible to represent any epimorphism \(\varphi:\pi_1(M)\to F_r\) as the Reeb epimorphism of a Morse function on \(M\). To construct the Morse function, the authors study a correspondence between epimorphisms \(\varphi:\pi_1(M)\to F_r\) and systems of \(r\) framed non-separating hypersurfaces in \(M\), and show that this correspondence induces a bijection onto framed cobordism classes of such systems. The hypersurfaces can be considered as connected components of some level sets of the constructed Morse function. This correspondence also provides a geometric tool for studying the general problem of classifying epimorphisms of groups \(G\to F_r\).\N\NSome applications to the Reeb graph theory are also presented, including consideration of manifolds with boundary.