## TRANSLATION GROUPOIDS AND ORBIFOLD BREDON COHOMOLOGY (705)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Pronk705translationgroupoids,

author = {Dorette Pronk and Laura Scull},

title = {TRANSLATION GROUPOIDS AND ORBIFOLD BREDON COHOMOLOGY},

year = {705}

}

### OpenURL

### Abstract

Abstract. We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an example, we use this result to define an orbifold version of Bredon cohomology. 1.

### Citations

128 |
A new cohomology theory of orbifold
- Chen, Ruan
(Show Context)
Citation Context ...h common refinement is used; details can be found in [20, 21]. Note that the original definition required that all group actions be effective, but it has been shown in recent papers (see for example, =-=[3]-=- or [9]) that it is often useful to drop this requirement; we will not require that G acts effectively on Ũ. Working with orbifold atlases is cumbersome, particularly when dealing with maps between or... |

69 |
Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A
- Hilsum, Skandalis
- 1987
(Show Context)
Citation Context ...d also the corresponding replacements for the 2-cells), we make use of an alternate way of describing maps between orbifold Lie groupoids, based on groupoid bundles, developed by Hilsum and Skandalis =-=[8]-=-. Some background on Hilsum-Skandalis maps is given in Section 5, and Section 6 gives the proof of Theorem 3.4. Thus we can work with just the equivariant maps between translation groupoids, with the ... |

69 |
Equivariant homotopy and cohomology theory. With contributions by
- May
- 1996
(Show Context)
Citation Context ...paces with an action of the group G and equivariant maps between them. Much of ordinary homotopy theory can be adapted and extended to this setting, although there are some important differences; see =-=[10]-=- for an overview of this theory. From another point of view, there has been much recent interest in the study of orbifolds, which are something like manifolds but whose local structure is a quotient o... |

68 |
Equivariant cohomology theories
- Bredon
- 1967
(Show Context)
Citation Context ...etween representable orbifolds and translation groupoids can be used to import equivariant information into the orbifold category. We define an orbifold version of the G-cohomology theories of Bredon =-=[2]-=- with constant coefficients (coefficients which do not depend on the space, only on the group G and its orbit category). These Bredon cohomology theories are defined for coefficients given by diagrams... |

48 |
Groupoïdes d’holonomie et classifiants, Astérisque
- Haefliger
- 1984
(Show Context)
Citation Context ...ering this category rather than its 2categorical refinement is that there is a nice description of the morphisms in terms of groupoid bundles. The resulting morphisms are called Hilsum-Skandalis maps =-=[5, 17]-=-. In [13], Moerdijk and Mrčun give a description of the correspondence between isomorphism classes of generalized maps and Hilsum-Skandalis maps, which we will use in the proof of Theorem 3.4. In this... |

40 |
Twisted orbifold K-theory
- Adem, Ruan
(Show Context)
Citation Context ... much recent interest in the study of orbifolds, which are something like manifolds but whose local structure is a quotient of an open subset of a Euclidean space by a finite group action ([20]; also =-=[1, 9]-=-). Although many of the basic geometric structures are the same in both cases, the techniques of these two approaches have been rather different. The goal of this paper is to provide a way of moving b... |

20 |
The classifying topos of a continuous groupoid
- Moerdijk
- 1988
(Show Context)
Citation Context ...st fibre product is again a submersion, so we obtain another manifold. It can also be shown that the fibre product of an essential equivalence along any homomorphism is again an essential equivalence =-=[11]-=-; thus any zig-zag of essential equivalences may be shortened by taking a fibre product, and so by repeated shortening, replaced by a single span as above. The class W of essential equivalences betwee... |

9 |
Presentations of Noneffective Orbifolds
- Henriques, Metzler
(Show Context)
Citation Context ... way to obtain an orbifold is to look at the action of a compact Lie group acting on a manifold with finite stabilizers. In fact, a large class (perhaps all) of orbifolds can be described in this way =-=[7]-=-, although this description is not unique for a given orbifold. Orbifolds that can be described this way are called representable. We can try to import equivariant invariants to these representable or... |

7 |
Gerbes over orbifolds and twisted
- Lupercio, Uribe
(Show Context)
Citation Context ... much recent interest in the study of orbifolds, which are something like manifolds but whose local structure is a quotient of an open subset of a Euclidean space by a finite group action ([20]; also =-=[1, 9]-=-). Although many of the basic geometric structures are the same in both cases, the techniques of these two approaches have been rather different. The goal of this paper is to provide a way of moving b... |

4 | orbifold quantum cohomology, Preprint AG/0005198 - Chen, Ruan |

3 |
Orbifolds as groupoids: an introduction Contemp
- Moerdijk
- 2002
(Show Context)
Citation Context ...arious representations of the same orbifold, so that our theory implies that this is indeed an orbifold invariant. This particular Bredon cohomology has been looked at in various other ways. Moerdijk =-=[12]-=- has used a Leray spectralTRANSLATION GROUPOIDS AND ORBIFOLD BREDON COHOMOLOGY 3 sequence argument to show that over the ring C of complex numbers, this particular Bredon cohomology can also be obtai... |

3 |
Lie groupoids, sheaves and cohomology
- Moerdijk, Mrčun
- 2005
(Show Context)
Citation Context ...ategory rather than its 2categorical refinement is that there is a nice description of the morphisms in terms of groupoid bundles. The resulting morphisms are called Hilsum-Skandalis maps [5, 17]. In =-=[13]-=-, Moerdijk and Mrčun give a description of the correspondence between isomorphism classes of generalized maps and Hilsum-Skandalis maps, which we will use in the proof of Theorem 3.4. In this section,... |

1 |
Equivariant Alexander-Spanier cohomology for actions of compact Lie groups
- Honkasalo
- 1990
(Show Context)
Citation Context ...ls Q. The orbifold Bredon cohomology we define can also be derived from a description of the Bredon cohomology in terms of the sheaf cohomology of the underlying quotient space, given by Honkasalo in =-=[6]-=-, although he did not consider the possibility of applying his work to orbifolds. Our approach gives a clearer idea of the relationship between the equivariant and orbifold phenomena, and is a bluepri... |