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Translation Groupoids and Orbifold Cohomology

Published online by Cambridge University Press:  20 November 2018

Dorette Pronk  and
Laura Scull
Dorette Pronk
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, e-mail: [email protected]
Laura Scull
Affiliation:
Department of Mathematics, Fort Lewis College, Durango, CO 81301-3999, USA, e-mail: scull [email protected]
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Abstract

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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.

Keywords

57S1555N9119L4718D0518D35orbifoldsequivariant homotopy theorytranslation groupoidsbicategories of fractions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 3 , 01 June 2010 , pp. 614 - 645
Copyright
Copyright © Canadian Mathematical Society 2010

References

[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] Adem, A. and Ruan, Y., Twisted orbifold K-theory. Comm. Math. Phys. 237(2003), no. 3, 533–556.Google Scholar
[3] Bredon, G., Equivariant Cohomology Theories. Lecture Notes in Mathematics 34. Springer-Verlag, Berlin, 1967.Google Scholar
[4] Chen, W. and Ruan, Y., A new cohomology theory of orbifold. Comm. Math. Phys. 248(2004), no.1, 1–31.Google Scholar
[5] Haefliger, A., Groupöıdes d’holonomie et classifiants. Astérisque No. 116(1984), 70–97.Google Scholar
[6] Honkasalo, H., Equivariant Alexander-Spanier cohomology for actions of compact Lie groups. Math. Scand. 67(1990), no. 1, 23–34.Google Scholar
[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] 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] 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] 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] 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] 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] 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] 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] 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] 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] Pradines, J., Morphisms between spaces of leaves viewed as fractions. Cahiers Topologie Géom. Différentielle Catég. 30(1989), no. 3, 229–246.Google Scholar
[18] Pronk, D. A., Groupoid Representations for Sheaves on Orbifolds, Ph.D. thesis, Utrecht, 1995.Google Scholar
[19] Pronk, D. A., Etendues and stacks as bicategories of factions. Compositio Math. 102(1996), no. 3, 243–303.Google Scholar
[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] Satake, I., The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan 9(1957), 464–492.Google Scholar
[22] Segal, G. B., Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math. No. 34(1968), 129–151.Google Scholar
[23] Willson, S. J., Equivariant homology theories on G-complexes. Trans. Amer. Math. Soc. 212(1975), 155–271. doi:10.2307/1998619 Google Scholar