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An orbifold is much like a smooth manifold but possibly with singularities of the form of fixed points of group-actions.

Where a smooth manifold is a space locally modeled on Cartesian spaces/Euclidean spaces n\mathbb{R}^n, an orbifold is, more generally, a space that is locally modeled on smooth action groupoids (homotopy quotients) nG\mathbb{R}^n\sslash G of a finite group GG acting on a Cartesian space.

graphics grabbed from Hyde-Ramsden-Robins 14

This turns out to be broadly captured(Moerdijk-Pronk 97, Moerdijk 02) by saying that an orbifold is a proper étale Lie groupoid. (Morita equivalent Lie groupoids correspond to the same orbifolds.)

The word orbifold was invented in (Thurston 1992), while the original name was VV-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowadays we call such orbifolds effective and those which are global quotients by a finite group global quotient orbifolds.

There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities.


An orbifold is a stack presented by an orbifold groupoid.



Global quotient orbifolds

In (ALR 07, theorem 1.23) is asserted that every effective orbifold XX (paracompact, Hausdorff) is isomorphic to a global quotient orbifold, specifically to a global quotient of O(n)O(n) (where nn is the dimension of XX) acting on the frame bundle of XX (which is a manifold).


It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to orbifold Euler characteristics, orbifold cohomology etc. One of the constructions which is useful in this respect is the inertia orbifold (the inertia stack of the original orbifold) which gives rise to “twisted sectors” in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.


Orbifolds are in differential geometry what Deligne-Mumford stacks are in algebraic geometry. See also at geometric invariant theory and GIT-stable point.

If the finiteness condition is dropped one also speaks of orbispaces and generally of stacks.

Orbifolds may be regarded as a kind of stratified spaces.

See also

Orbifolds in string theory:



The concept originates in

  • I. Satake, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363.

  • I. Satake, The Gauss–Bonnet theorem for VV-manifolds, J. Math. Soc. Japan 9 (1957), 464–492.

  • 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; in particular the orbifold discussion is in chapter 13.

Survey of basic orbifold theory:

  • Daryl Cooper, Craig Hodgson, Steve Kerckhoff, Three-dimensional Orbifolds and Cone-Manifolds, MSJ Memoirs Volume 5, 2000 (pdf, euclid:1389985812)

  • Adam Kaye, Two-Dimensional Orbifolds, 2007 (pdf)

  • Michael Davis, Lectures on orbifolds and reflection groups, 2008 (pdf)

  • Joan Porti, An introduction to orbifolds, 2009 (pdf)

On Riemannian orbifolds:

  • Christian Lange, Orbifolds from a metric viewpoint (arXiv:1801.03472)

  • Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, Teichmüller theory and collapse of flat manifolds, Annali di Matematica (2018) 197: 1247 (arXiv:1705.08431, doi:10.1007/s10231-017-0723-7)

  • S. T. Hyde, S. J. Ramsden and V. Robins, Unification and classification of two-dimensional crystalline patterns using orbifolds, Acta Cryst. (2014). A70, 319-337 (doi:10.1107/S205327331400549X)

Discussion of orbifold as Lie groupoids/differentiable stacks is in

The mapping stacks of orbifolds are discussed in

  • W. Chen, On a notion of maps between orbifolds, I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569–620.

Orbifolds often appear as moduli spaces in differential geometric setting:

The generalization of orbifolds to weighted branched manifolds is discussed in

  • Dusa McDuff, Groupoids, branched manifolds and multisections, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (project euclid).

A reference dealing with the string topology of orbifolds is

  • A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (pdf)

The relation of orbifolds to global equivariant homotopy theory is discussed in

See also

(which is mainly tailored toward Thurston’s approach).

Orbifold cobordism

Orbifold cobordisms are discussed in

  • K. S. Druschel, Oriented Orbifold Cobordism, Pacific J. Math., 164(2) (1994), 299-319.

  • K. S. Druschel, The Cobordism of Oriented Three Dimensional Orbifolds, Pacific J. Math., bf 193(1) (2000), 45-55.

  • Andres Angel, Orbifold cobordism (pdf)

See also at orbifold cobordism.

In string theory

In perturbative string theory, orbifolds as target spaces for a string sigma-model were first considered in

and then further developed notably in

See also the references at fractional D-brane.

Review of orbifolds in the context of string compactifications includes

and for orbifolds of G2-manifolds for M-theory on G2-manifolds

For topological strings the path integral as a pull-push transform for target orbifolds – in analogy to what Gromov-Witten theory is for Deligne-Mumford stacks – has first been considered in

  • Weimin Chen, Yongbin Ruan, Orbifold Gromov-Witten Theory, in Orbifolds in mathematics and physics (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (arXiv:math/0103156)

A review with further pointers is in

category: Lie theory

Last revised on December 12, 2018 at 11:50:58. See the history of this page for a list of all contributions to it.