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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
DOI: https://doi.org/10.1090/S0002-9947-04-03379-3
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.


References [Enhancements On Off] (What's this?)

  • [AR] Alejandro Adem and Yongbin Ruan.
    Twisted Orbifold K-Theory.
    Contemporary Mathematics, 310. American Mathematical Society, Providence, RI, 2002. MR 2003g:00020
  • [Bry93] Jean-Luc Brylinski.
    Loop spaces, characteristic classes and geometric quantization, volume 107 of Progress in Mathematics.
    Birkhäuser Boston Inc., Boston, MA, 1993. MR 94b:57030
  • [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(4):319-362, 1999. MR 2000j:19002
  • [EHKV] D. Edidin, B. Hassett, A. Kresch, and A. Vistoli.
    Brauer groups and quotient stacks. Amer. J. Math. 123(4):761-777, 2001. MR 2002f:14002
  • [Gir71] Jean Giraud.
    Cohomologie non abélienne.
    Springer-Verlag, Berlin, 1971.
    Die Grundlehren der mathematischen Wissenschaften, Band 179. MR 49:8992
  • [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, 17:75-83, 1978. MR 57:14072
  • [Kaw79] Tetsuro Kawasaki.
    The Riemann-Roch theorem for complex V-manifolds.
    Osaka J. Math., 16:151-159, 1979. MR 80f:58042
  • [LMB00] Gérard Laumon and Laurent Moret-Bailly.
    Champs algébriques, volume 39 of 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].
    Springer-Verlag, Berlin, 2000. MR 2001f:14006
  • [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(1):3-21, 1997. MR 98i:22004
  • [Pro96] Dorette A. Pronk.
    Etendues and stacks as bicategories of fractions.
    Compositio Math., 102(3):243-303, 1996. MR 97d:18011
  • [Rua] Yongbin Ruan.
    Stringy Geometry and Topology of Orbifolds.
    Preprint, available as arXiv:math.AG/0011149.
  • [Sat56] Ichiro Satake.
    On a generalization of the notion of manifold.
    Proc. Nat. Acad. Sci. USA, 42:359-363, 1956. MR 18:144a

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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: https://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

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