Geodesic
Translation Groupoids and Orbifold Cohomology
Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 614-645

Voir la notice de l'article provenant de la source Cambridge University Press

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 application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: $K$ -theory and Bredon cohomology for certain coefficient diagrams.
DOI : 10.4153/CJM-2010-024-1
Mots-clés : 57S15, 55N91, 19L47, 18D05, 18D35, orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions 
Pronk, Dorette; Scull, Laura. Translation Groupoids and Orbifold Cohomology. Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 614-645. doi: 10.4153/CJM-2010-024-1
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[1] [1] Adem, A., Leida, J., and Ruan, Y., Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics 171, Cambridge University Press, Cambridge 2007. Google Scholar

[2] [2] Adem, A. and Ruan, Y., Twisted orbifold K-theory. Comm. Math. Phys. 237(2003), no. 3, 533–556. Google Scholar

[3] [3] Bredon, G., Equivariant Cohomology Theories. Lecture Notes in Mathematics 34. Springer-Verlag, Berlin, 1967. Google Scholar

[4] [4] Chen, W. and Ruan, Y., A new cohomology theory of orbifold. Comm. Math. Phys. 248(2004), no.1, 1–31. Google Scholar

[5] [5] Haefliger, A., Groupöıdes d’holonomie et classifiants. Astérisque No. 116(1984), 70–97. Google Scholar

[6] [6] Honkasalo, H., Equivariant Alexander-Spanier cohomology for actions of compact Lie groups. Math. Scand. 67(1990), no. 1, 23–34. Google Scholar

[7] [7] Henriques, A. and Metzler, D., Presentations of noneffective orbifolds. Trans. Amer. Math. Soc. 356(2004), no. 6, 2481–2499. doi: 10.1090/S0002-9947-04-03379-3. Google Scholar

[8] [8] Hilsum, M. and Skandalis, G., Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’aprés une conjecture d’A. Connes). Ann. Sci. École Norm. Sup. 20(1987), no. 3, 325–390. Google Scholar

[9] [9] Lupercio, E. and Uribe, B., Gerbes over orbifolds and twisted K-theory,. Comm. Math. Phys. 245(2004), no. 3, 449–489. doi: 10.1007/s00220-003-1035-x Google Scholar

[10] [10] May, J. P., Equivariant Homotopy and Cohomology Theory. CB MS Regional Conference Series in Mathematics 91. American Mathematical Society, Providence, RI, 1996. Google Scholar

[11] [11] Moerdijk, I., The classifying topos of a continuous groupoid. I. Trans. Amer. Math. Soc. 310(1988), no. 2, 629–668. doi: 10.2307/2000984 Google Scholar

[12] [12] Moerdijk, I., Orbifolds as groupoids: an introduction. In: Orbifolds in Mathematics and Physics. Contemp. Math. 310. American Mathematical Society, Providence, RI, 2002, pp. 205–222. Google Scholar

[13] [13] Moerdijk, I. and Mrčun, J., Lie groupoids, sheaves and cohomology. In: Poisson Geometry, Deformation Quantisation and Group Representations. London Math. Soc. Lecture Note Ser. 323. Cambridge University Press, Cambridge, 2005, pp. 145–272. Google Scholar

[14] [14] Moerdijk, I., and Pronk, D. A., Orbifolds, sheaves and groupoids. K-Theory 12(1997), no. 1, 3–21. doi: 10.1023/A:1007767628271 Google Scholar

[15] [15] Moerdijk, I., and Svensson, J. A., A Shapiro lemma for diagrams of spaces with applications to equivariant topology. Compositio Math. 96(1995), no. 3, 249–282. Google Scholar

[16] [16] Mrčun, J., Functoriality of the bimodule associated to a Hilsum-Skandalis map. K-Theory 18(1999),no. 3, 235–253. doi: 10.1023/A:1007773511327 Google Scholar

[17] [17] Pradines, J., Morphisms between spaces of leaves viewed as fractions. Cahiers Topologie Géom. Différentielle Catég. 30(1989), no. 3, 229–246. Google Scholar

[18] [18] Pronk, D. A., Groupoid Representations for Sheaves on Orbifolds, Ph.D. thesis, Utrecht, 1995. Google Scholar

[19] [19] Pronk, D. A., Etendues and stacks as bicategories of factions. Compositio Math. 102(1996), no. 3, 243–303. Google Scholar

[20] [20] Satake, I., On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. U.S.A. 42(1956), 359–363. doi: 10.1073/pnas.42.6.359 Google Scholar

[21] [21] Satake, I., The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9(1957), 464–492. Google Scholar

[22] [22] Segal, G. B., Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math. No. 34(1968), 129–151. Google Scholar

[23] [23] Willson, S. J., Equivariant homology theories on G-complexes. Trans. Amer. Math. Soc. 212(1975), 155–271. doi: 10.2307/1998619 Google Scholar

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