Reeb graph invariants of Morse functions and 3-manifold groups

2024/03/04 by Łukasz Patryk Michalak, Michalak, Łukasz Patryk
Computer Science · Mathematics · #20F05 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Primary: 57M15 #Secondary: 57K31 #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2403.02291

openalex publication_date 2024/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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(π₁(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.

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