higher geometry / derived geometry
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∞-Lie theory (higher geometry)
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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 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: (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
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)
Created on November 4, 2021 at 10:30:03. See the history of this page for a list of all contributions to it.