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Presentations of noneffective orbifolds


Authors: Andre Henriques and David S. Metzler
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2481-2499
MSC (2000): Primary 58H05; Secondary 57S10, 18F99
Published electronically: February 2, 2004
MathSciNet review: 2048526
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Abstract: It is well known that an effective orbifold $M$ (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold $P$ by a locally free action of a compact Lie group $K$. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.


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  • [AR] Alejandro Adem, Jack Morava, and Yongbin Ruan (eds.), Orbifolds in mathematics and physics, Contemporary Mathematics, vol. 310, American Mathematical Society, Providence, RI, 2002. MR 1950939
  • [Bry93] Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1197353
  • [CR] Weimin Chen and Yongbin Ruan.
    A New Cohomology Theory for Orbifold.
    Preprint, available as arXiv:math.AG/0004129.
  • [Cra99] Marius Crainic, Cyclic cohomology of étale groupoids: the general case, 𝐾-Theory 17 (1999), no. 4, 319–362. MR 1706117, 10.1023/A:1007756702025
  • [EHKV] Dan Edidin, Brendan Hassett, Andrew Kresch, and Angelo Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), no. 4, 761–777. MR 1844577
  • [Gir71] Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin-New York, 1971 (French). Die Grundlehren der mathematischen Wissenschaften, Band 179. MR 0344253
  • [Gro68] Alexander Grothendieck.
    Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses.
    In Dix Exposés sur la Cohomologie des Schémas, pages 46-66. North-Holland, Amsterdam, 1968. MR 39:5586a
  • [Hen] Andre Henriques.
    Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition.
  • [Kaw78] Tetsuro Kawasaki, The signature theorem for 𝑉-manifolds, Topology 17 (1978), no. 1, 75–83. MR 0474432
  • [Kaw79] Tetsuro Kawasaki, The Riemann-Roch theorem for complex 𝑉-manifolds, Osaka J. Math. 16 (1979), no. 1, 151–159. MR 527023
  • [LMB00] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
  • [LU] Ernesto Lupercio and Bernardo Uribe.
    Gerbes over Orbifolds and Twisted K-theory.
    Preprint, available as arXiv:math.AT/0105039.
  • [Met] David Metzler.
    Topological stacks, gerbes, groupoids, and orbispaces.
    in preparation.
  • [Moe] Ieke Moerdijk.
    Orbifolds as Groupoids: an Introduction.
    Orbifolds in mathematics and physics (Madison, WI, 2001), 205-222, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002.
  • [MP97] I. Moerdijk and D. A. Pronk, Orbifolds, sheaves and groupoids, 𝐾-Theory 12 (1997), no. 1, 3–21. MR 1466622, 10.1023/A:1007767628271
  • [Pro96] Dorette A. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), no. 3, 243–303. MR 1401424
  • [Rua] Yongbin Ruan.
    Stringy Geometry and Topology of Orbifolds.
    Preprint, available as arXiv:math.AG/0011149.
  • [Sat56] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. MR 0079769

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Additional Information

Andre Henriques
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: andrhenr@mit.edu

David S. Metzler
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: metzler@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03379-3
Keywords: Orbifolds, groupoids, stacks, gerbes, group actions
Received by editor(s): February 12, 2003
Received by editor(s) in revised form: April 29, 2003
Published electronically: February 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society