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

Wacław Marzantowicz, Łukasz Patryk Michalak

Published 2020-02-06Version 1

In this work we present a construction which gives a correspondence between epimorphisms $\varphi \colon \pi_1(W) \to F_r$, from the fundamental group of a compact manifold $W$ onto the free group of rank $r$, and systems of framed non-separating hypersurfaces in $W$. In consequence, any such $\varphi$, which corresponds to a system of hypersurfaces without boundary, can be represented by the Reeb epimorphism of a Morse function $f\colon W \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $W \to \mathcal{R}(f)$ onto the Reeb graph of $f$. We study properties and natural relations between these three objects. In particular, from this point of view we discuss the problem of classification up to (strong-)equivalence of epimorphisms onto free groups and we provide a purely geometrical-topological proof of the solution of this problem for surface groups which was given earlier by Grigorchuk, Kurchanov and Zieshang by using other methods.