nLab
presentable orbifold

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

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

An orbifold is called presentable if it is (Morita-)equivalently a global quotient orbifold of a smooth manifold by the action of some Lie group (sometimes assumed/required to be compact).

For example, (very) good orbifolds are presentable, since (finite) discrete groups are examples of (compact) Lie groups.

More interestingly, every effective orbifold 𝒳\mathcal{X} is presentable, namely as the quotient of its frame bundle by the general linear group, or equivalently of its orthonormal frame bundle (with respect to some Riemannian structure) by the orthogonal group: 𝒳Fr orth(𝒳)O(dim(𝒳))\mathcal{X} \,\simeq\, Fr_{orth}(\mathcal{X}) \sslash O(dim(\mathcal{X})) (Satake 1956, recalled in, e.g., Henriques & Metzler 2004, Prop. 1.4).

In fact, it is conjectured that every orbifold is presentable as the global quotient of a smooth manifold by some Lie group (Adem, Leida & Ruan 2007, Conj. 1.55). Some results in this direction are presented in Henriques & Metzler 2004, Sec. 5

References