nLab good orbifold

Contents

Context

Ingredients

Concepts

Constructions

Examples

Theorems

Higher Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

Contents

Definition

An orbifold $\mathcal{X}$ is called good (Thurston 92, Ch. 13, Def. 13.2.3) or developable if it is isomorphic to a global quotient of a smooth manifold $M$ by the action of a discrete group $\flat G$ (not necessarily finite):

$\mathcal{X} \,\simeq\, M \sslash \flat G$

Otherwise $\mathcal{X}$ is called bad.

If $\mathcal{X}$ is even the global quotient of a smooth manifold by a finite group action one says that it is very good.

In the other direction, an orbifold that is the global quotient of a smooth manifold by some (compact) Lie group is called a presentable orbifold.

References

The first three chapters are published in expanded form as