nLab presentable orbifold

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 $\mathcal{X}$ 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: $\mathcal{X} \,\simeq\, Fr_{orth}(\mathcal{X}) \sslash O(dim(\mathcal{X}))$ (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

• Ichiro Satake, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 (doi:10.1073/pnas.42.6.359)

• André Henriques, David Metzler, Presentations of Noneffective Orbifolds, Trans. Amer. Math. Soc. 356 (2004), 2481-2499 (arXiv:math/0302182, doi:10.1090/S0002-9947-04-03379-3)

• Alejandro Adem, Johann Leida, Yongbin Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (doi:10.1017/CBO9780511543081, pdf)