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In this work we present a construction of correspondence between epimorphisms$\varphi \colon \pi_1(M) \to F_r$ from the fundamental group of a compactmanifold $M$ onto the free group of rank $r$, and systems of $r$ framednon-separating hypersurfaces in $M$, which induces a bijection onto theirframed cobordism classes. In consequence, for closed manifolds any such$\varphi$ can be represented by the Reeb epimorphism of a Morse function$f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map$M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this constructionwe discuss the problem of classification up to (strong) equivalence ofepimorphisms onto free groups providing a new purely geometrical-topologicalproof of the solution of this problem for surface groups.

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