## TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY (2003)

### BibTeX

@MISC{Henriques03transactionsof,

author = {Andre Henriques and David and S. Metzler},

title = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY},

year = {2003}

}

### OpenURL

### Abstract

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. 1.

### Citations

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Cohomologie non abélienne
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- 1971
(Show Context)
Citation Context ...e extra structure is a well-known one, at least to algebraic geometers. Such a groupoid corresponds to a gerbe on the underlying manifold Gtop. We recall the definition of a gerbe F on a space M. See =-=[Gir71]-=-, [Bry93], [LMB00] for more details. A gerbe is roughly a certain kind of “sheaf of categories.” (More precisely, it is a certain kind of stack.) Definition 4.4. Let M be a topological space. A (stric... |

43 | Stringy geometry and topology of orbifolds
- Ruan
- 2000
(Show Context)
Citation Context ...o note some connections to stacks and gerbes. 1. Introduction Recently, motivated largely by string theory, many researchers have developed new aspects of the geometry and topology of orbifolds (e.g. =-=[Rua]-=-, [CR], [AR]). In particular, it has become clear that the traditional definition (originally due to Satake [Sat56]) is cumbersome and sometimes misleading. Orbifolds should not be seen simply as mani... |

40 |
Twisted orbifold K-theory
- Adem, Ruan
(Show Context)
Citation Context ...onnections to stacks and gerbes. 1. Introduction Recently, motivated largely by string theory, many researchers have developed new aspects of the geometry and topology of orbifolds (e.g. [Rua], [CR], =-=[AR]-=-). In particular, it has become clear that the traditional definition (originally due to Satake [Sat56]) is cumbersome and sometimes misleading. Orbifolds should not be seen simply as manifolds with m... |

38 |
Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A
- Laumon, Moret-Bailly
- 2000
(Show Context)
Citation Context ...is a well-known one, at least to algebraic geometers. Such a groupoid corresponds to a gerbe on the underlying manifold Gtop. We recall the definition of a gerbe F on a space M. See [Gir71], [Bry93], =-=[LMB00]-=- for more details. A gerbe is roughly a certain kind of “sheaf of categories.” (More precisely, it is a certain kind of stack.) Definition 4.4. Let M be a topological space. A (strict) gerbe F on M is... |

33 |
spaces, characteristic classes and geometric quantization, volume 107
- Loop
- 1993
(Show Context)
Citation Context ...tructure is a well-known one, at least to algebraic geometers. Such a groupoid corresponds to a gerbe on the underlying manifold Gtop. We recall the definition of a gerbe F on a space M. See [Gir71], =-=[Bry93]-=-, [LMB00] for more details. A gerbe is roughly a certain kind of “sheaf of categories.” (More precisely, it is a certain kind of stack.) Definition 4.4. Let M be a topological space. A (strict) gerbe ... |

32 | Orbifolds as groupoids: an introduction
- Moerdijk
- 2002
(Show Context)
Citation Context ...ly review the notion of groupoid, and introduce the structures on groupoids which we will need in the sequel. A good source for some of the material in this section is the recent preprint of Moerdijk =-=[Moe]-=-. In Section 2.4 we will address how to think of orbifolds as groupoids. A groupoid is a category where all morphisms are invertible. We need the notion of a smooth groupoid, where the sets of objects... |

21 | Cyclic cohomology of étale groupoids: the general case, K-theory 17
- Crainic
- 1999
(Show Context)
Citation Context ...ids. Since we are only interested in G as a model for its quotient, we need to be looser about what a morphism can be. The appropriate notion of map is that of Hilsum and Skandalis. We follow Crainic =-=[Cra99]-=- here. The key is that we are (roughly) allowed to “replace” the source groupoid by an “equivalent” object. Here the equivalent object is a principal bundle over the space of objects of the source. De... |

20 |
The Riemann-Roch theorem for complex V-manifolds
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- 1979
(Show Context)
Citation Context ... wants to utilize the extra structure associated to the stabilizer groups, this restriction becomes artificial. In addition, certain useful constructions on orbifolds (such as Kawasaki’s ˜M, [Kaw78], =-=[Kaw79]-=-) take one out of the effective situation. Even the extreme case where the actions of the stabilizer groups are all trivial turns out to be interesting; the presentation problem here is the topologica... |

19 |
Toposes and groupoids
- Moerdijk
- 1988
(Show Context)
Citation Context ...ly in the orbifold literature, for example by Ruan [Rua], who noted that to pull back an “orbi-vector bundle,” one needs more than a map on the underlying topological space, and by Moerdijk and Pronk =-=[MP97]-=-, who observed the same thing for sheaves. Also see [Hen] for concrete examples showing the need for more data. This paper uses the groupoid language to (partially) answer a question about the present... |

16 |
RuanA new Cohomology Theory of orbifold, preprint
- Chen, Y
- 2000
(Show Context)
Citation Context ...some connections to stacks and gerbes. 1. Introduction Recently, motivated largely by string theory, many researchers have developed new aspects of the geometry and topology of orbifolds (e.g. [Rua], =-=[CR]-=-, [AR]). In particular, it has become clear that the traditional definition (originally due to Satake [Sat56]) is cumbersome and sometimes misleading. Orbifolds should not be seen simply as manifolds ... |

16 |
Etendues and stacks as bicategories of fractions
- Pronk
- 1996
(Show Context)
Citation Context ...ntial equivalences (in the sense of localization of categories). However we prefer the HS definition, partly because it leads more easily to the “stacky” nature of groupoids, see Section 3. (See also =-=[Pro96]-=-.)PRESENTATIONS OF NONEFFECTIVE ORBIFOLDS 9 Example. We can let the source and the target of an HS morphism be translation groupoids. Let X be a K-space, Y be an L-space, for Lie groups K, L. Lemma 2... |

16 |
On a generalisation of the notion of
- Satake
- 1956
(Show Context)
Citation Context ...researchers have developed new aspects of the geometry and topology of orbifolds (e.g. [Rua], [CR], [AR]). In particular, it has become clear that the traditional definition (originally due to Satake =-=[Sat56]-=-) is cumbersome and sometimes misleading. Orbifolds should not be seen simply as manifolds with mild singularities (i.e. a very nice kind of stratified space). Instead one wants to take into account t... |

12 |
Algèbres d’Azumaya et interprétations diverses [ MR0244269 (39 #5586a
- I
- 1995
(Show Context)
Citation Context ...sponding central embedding in any U(nk), and hence an embedding i∞ : A → U(∞). We have an exact sequence 0 → A → U(∞) → U(∞)/A → 0. The following result makes more precise a result of Serre, cited in =-=[Gro68]-=-. Proposition 5.8. Let A be a cyclic group and let S ⊂ K(A, 2) be an m-dimensional skeleton. Then there is an integer n = nm and a central embedding i : A → U(n) such that the associated Bockstein map... |

12 |
The signature theorem for V-manifold
- Kawasaki
- 1978
(Show Context)
Citation Context ... when one wants to utilize the extra structure associated to the stabilizer groups, this restriction becomes artificial. In addition, certain useful constructions on orbifolds (such as Kawasaki’s ˜M, =-=[Kaw78]-=-, [Kaw79]) take one out of the effective situation. Even the extreme case where the actions of the stabilizer groups are all trivial turns out to be interesting; the presentation problem here is the t... |

4 |
Vistoli: Brauer groups and quotient stacks. math.AG/9905049
- Edidin, Hassett, et al.
(Show Context)
Citation Context .... Even the extreme case where the actions of the stabilizer groups are all trivial turns out to be interesting; the presentation problem here is the topological analogue of the Brauer Conjecture (see =-=[EHKV]-=-) in algebraic geometry. We will see that the appropriate language for this case uses gerbes. Hence a natural question arises, can a noneffective orbifold always be presented? The answer is still unkn... |

2 | Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition
- Henriques
(Show Context)
Citation Context ...who noted that to pull back an “orbi-vector bundle,” one needs more than a map on the underlying topological space, and by Moerdijk and Pronk [MP97], who observed the same thing for sheaves. Also see =-=[Hen]-=- for concrete examples showing the need for more data. This paper uses the groupoid language to (partially) answer a question about the presentation of orbifolds. Orbifolds often arise as a quotient o... |

1 |
Gerbes over Orbifolds and Twisted K-theory. Preprint, available as arXiv:math.AT/0105039
- Lupercio, Uribe
(Show Context)
Citation Context ...tage of the extra structure of an orbifold—for example, to do “stringy geometry”—one must, one way or another, look at the stack structure on the orbifold. Lupercio and Uribe have pointed this out in =-=[LU]-=-, although they do not develop it. More concretely, stacks are represented by groupoids, and it is in this language that many constructions are easier than in the traditional orbifold language. In par... |

1 |
Topological stacks, gerbes, groupoids, and orbispaces
- Metzler
(Show Context)
Citation Context ...age, this says that a groupoid G naturally induces a stack on the category of manifolds. We present some further explanation of this idea and clarification of some confusing issues in a related paper =-=[Met]-=-. 4. Gerbes We will be particularly interested in a very simple kind of orbifold groupoid G, namely one whose corresponding effective orbifold Geff is equivalent to a manifold. Definition 4.1. A smoot... |