nLab good orbifold



Higher geometry

Higher Lie theory

∞-Lie theory (higher geometry)


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Higher groupoids

Lie theory

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\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



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.


  • William Thurston, Three-dimensional geometry and topology, preliminary draft, University of Minnesota, Minnesota, (1992)

    which in completed and revised form is available as his book:

    The Geometry and Topology of Three-Manifolds; (web)

    in particular the orbifold discussion is in chapter 13

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