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

We construct a correspondence between epimorphisms $\varphi \colon \pi _1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. 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 construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups.