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Combinatorial modifications of Reeb graphs and the realization problem

Łukasz Patryk Michalak

TL;DR

It is proved that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the number of cycles can be realized as the Reeb graph of a Morse function on a given closed manifold.

Abstract

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold $M$. Along the way, we show that the Reeb number $\mathcal{R}(M)$, i.e. the maximum cycle rank among all Reeb graphs of functions on $M$, is equal to the corank of fundamental group $π_1(M)$, thus extending a previous result of Gelbukh to the non-orientable case.

Combinatorial modifications of Reeb graphs and the realization problem

TL;DR

It is proved that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the number of cycles can be realized as the Reeb graph of a Morse function on a given closed manifold.

Abstract

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold . Along the way, we show that the Reeb number , i.e. the maximum cycle rank among all Reeb graphs of functions on , is equal to the corank of fundamental group , thus extending a previous result of Gelbukh to the non-orientable case.

Paper Structure

This paper contains 7 sections, 14 theorems, 16 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Index and degree correspondence
  4. Combinatorial modifications of Reeb graphs
  5. Reeb number and corank of fundamental group
  6. Realization theorem
  7. Further directions

Key Result

Proposition 3.1

Let $f\colon W \rightarrow \mathbb{R}$ be a simple Morse function on an $n$-dimensional smooth triad $(W,W_-,W_+)$, $n \geq 3$. Let $p$ be a critical point of $f$ and $v:= q(p)$ be the vertex in $\mathcal{R}(f)$ which corresponds to $p$. Then

Figures (13)

  • Figure 1: Example of the Reeb graph of height function $f\colon \operatorname{S}^1\times\operatorname{S}^1 \to \mathbb{R}$ on two-dimensional torus. Four critical points of $f$ have different values and correspond to four vertices in $\mathcal{R}(f)$.
  • Figure 2: (a) Graph not admitting a good orientation Sharko and (b) graph admitting it.
  • Figure 3: Possible neighbourhoods of a vertex of degree $3$ in the Reeb graph of a simple Morse function on a manifold of dimension at least $3$. We use our convention that the orientation is from the bottom to the top.
  • Figure 4: Combinatorial modifications of Reeb graphs related to change of the order of two consecutive critical points. On the right sides of vertices are their indices, where $1 \leq k \leq m \leq n-1$.
  • Figure 5: Combinatorial modifications of Reeb graphs related to Cancellation of Handles.
  • ...and 8 more figures

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 3.1: Reeb
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 33 more