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Reeb graph invariants of Morse functions and $3$-manifold groups

Łukasz Patryk Michalak

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 $\operatorname{corank}(π_1(M))$. 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.

Reeb graph invariants of Morse functions and $3$-manifold groups

Abstract

In this work we are focused on the existence of Morse functions on a closed manifold which are far from being ordered, i.e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals . In the case of -manifolds we describe the minimal number of critical points needed to construct such functions, which is related with the number of vertices of degree in Reeb graphs. We define a new invariant of -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.
Paper Structure (8 sections, 13 theorems, 32 equations, 1 figure)

This paper contains 8 sections, 13 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds on the Number of Degree $2$ Vertices
  4. Simple Morse Functions on $3$-Manifolds
  5. Low values of $\Delta_2(M)$
  6. Group presentation invariant
  7. Circle bundles over an orientable surface
  8. Final remarks

Key Result

Theorem 1.1

Let $M_{\pm 1}$ be the $\operatorname{S}^1$-bundle over an orientable surface $\Sigma_g$ of genus $g\geq 1$ with Euler number $e= \pm1$. Then $\operatorname{corank}(\pi_1(M_{\pm 1}))=g$ and Consequently, the Reeb graph of any simple Morse function $f\colon M_{\pm 1} \to \mathbb{R}$ with the minimum number of critical points is a tree, i.e. $\beta_1(\mathcal{R}(f)) = 0$, and $\Delta_2(\mathcal{R}(

Figures (1)

  • Figure 1: Genus $2g+1$ Heegaard splitting of $\operatorname{S}^1$-bundle $M_e$ over $\Sigma_g$ for $e \leq 0$. The same numbers at the points where $\gamma$ intersects the circles $A^{\pm}_i$ and $B^{\pm}_i$ indicate the identified points.

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • ...and 22 more