Reeb graph invariants of Morse functions and 3-manifold groups

In this work we are focused on the existence of Morse functions on a closed manifold M which are far from being ordered, i. e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals corank (₁ (M) ). In the case of 3-manifolds we describe the minimal number of critical points needed to construct such functions, which is related with the number of vertices of degree 2 in Reeb graphs. We define a new invariant of 3-manifold groups and their presentations, and using Heegaard splittings we show its utility in determining occurrence of disordered Morse functions. In particular, the Freiheitssatz, a result for one-relator groups, allows us to calculate this invariant in the case of orientable circle-bundles over a surface, which provides an interesting example of the behaviour of Morse functions.