nLab Reeb graph

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Given a topological space XX equipped with a continuous function f:X 1f \,\colon\, X \longrightarrow \mathbb{R}^1 to the Euclidean space of real numbers, it’s Reeb graph is the quotient topological space of XX by the equivalence relation which regards two points x,yXx, y \,\in\, X as equivalent iff they are in the same connected component of the same level set. Under good conditions this is a CW complex, necessarily 1-dimensional, and as such an undirected graph.

References

Original article:

  • Georges Reeb, Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique, Comptes Rendus Acad. Sciences Paris 222 (1946) 847-849 [[crid:1571417125676878592]]

Further developments:

See also:

Last revised on May 26, 2022 at 15:54:40. See the history of this page for a list of all contributions to it.