nLab good orbifold

Contents

Context

Higher geometry

Higher Lie theory

∞-Lie theory (higher geometry)

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 𝒳\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 MM by the action of a discrete group G\flat G (not necessarily finite):

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

Otherwise 𝒳\mathcal{X} is called bad.

If 𝒳\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 notion of good orbifolds is due to Def. 13.2.3 in:

Further discussion:

Last revised on June 8, 2024 at 13:05:03. See the history of this page for a list of all contributions to it.