Roughly, an orbifold is the quotientPlanetmathPlanetmath of a manifold by a finite groupMathworldPlanetmath. For example, take a sheet of paper and add a small crease perpendicular to one side at the halfway point. Then, line up the two halves of the side. This may be thought of as the plane 2 modulo the group 2. Now, let us give the definition.

Define a category 𝒳: The objects are pairs (G,X), where G is a finite group acting effectively on a connected smooth manifold X. A morphism Φ between two objects (G,X) and (G,X) is a family of open embeddingsMathworldPlanetmathPlanetmath ϕ:XX which satisfy

  • for each embedding ϕΦ, there is an injective homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath λϕ:GG such that ϕ is λϕ equivariant

  • For gG, we have

    gϕ :XX
    gϕ :xgϕ(x)

    and if (gϕ)(X)ϕ(X), then gλϕ(G).

  • Φ={gϕ,gG}, for any ϕΦ

Now, we define orbifolds. Given a paracompact Hausdorff space X and a nice open covering 𝒰 which forms a basis for the topologyMathworldPlanetmathPlanetmath on X, an orbifold structure 𝒱 on X consists of

  1. 1.

    For U𝒰, 𝒱(U)=(GU,U~)τU is a ramified cover U~U which identifies U~/GUU

  2. 2.

    For UV𝒰, there exists a morphism ϕVU(GU,U~)(GV,V~) covering the inclusion

  3. 3.

    If UVW𝒰, ϕWU=ϕWVϕVU

[1] Kawasaki T., The SignaturePlanetmathPlanetmathPlanetmath theoremMathworldPlanetmath for V-manifolds. Topology 17 (1978), 75-83. MR0474432 (57:14072)

Title orbifold
Canonical name Orbifold
Date of creation 2013-03-22 15:40:06
Last modified on 2013-03-22 15:40:06
Owner guffin (12505)
Last modified by guffin (12505)
Numerical id 8
Author guffin (12505)
Entry type Definition
Classification msc 57M07
Synonym orbifold structure
Defines orbifold structure