Twisted K-theory of differentiable stacks
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 37 (2004) no. 6, p. 841-910
@article{ASENS_2004_4_37_6_841_0,
     author = {Tu, Jean-Louis and Xu, Ping and Laurent-Gengoux, Camille},
     title = {Twisted $K$-theory of differentiable stacks},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 37},
     number = {6},
     year = {2004},
     pages = {841-910},
     doi = {10.1016/j.ansens.2004.10.002},
     zbl = {1069.19006},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2004_4_37_6_841_0}
}
Tu, Jean-Louis; Xu, Ping; Laurent-Gengoux, Camille. Twisted $K$-theory of differentiable stacks. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 37 (2004) no. 6, pp. 841-910. doi : 10.1016/j.ansens.2004.10.002. http://www.numdam.org/item/ASENS_2004_4_37_6_841_0/

[1] Adem A., Ruan Y., Twisted orbifold K-theory, Comm. Math. Phys. 237 (2003) 533-556. | MR 1993337 | Zbl 1051.57022

[2] Artin M. et al. , Théorie des topos et cohomologie étale des schémas, in: Séminaire de géométrie algébrique, Lecture Notes in Mathematics, vols. 269, 270, 305, Springer, Berlin, 1972-1973.

[3] Atiyah M., K-theory. Lecture notes by D.W. Anderson, 1967. | MR 224083

[4] Atiyah M., K-theory past and present, math.KT/0012213. | MR 2091892

[5] Atiyah M., Segal G., Twisted K-theory, math.KT/0407054.

[6] Baum P., Connes A., Chern character for discrete groups, in: A fête of topology, Academic Press, New York, 1988, pp. 163-232. | MR 928402 | Zbl 0656.55005

[7] Baum P., Connes A., Higson N., Classifying space for proper actions and K-theory of group C * -algebras, in: C * -Algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291. | MR 1292018 | Zbl 0830.46061

[8] Behrend K., Edidin B., Fantechi B., Fulton W., Gottsche L., Kresch K., Introduction to stacks, in preparation.

[9] Behrend K., Xu P., S 1 -bundles and gerbes over differentiable stack, C. R. Acad. Sci. Paris Sér. I 336 (2003) 163-168. | MR 1969572 | Zbl 1039.58016

[10] Behrend, K., Xu P., Differentiable stacks and gerbes, in preparation.

[11] Blackadar B., K-Theory for Operator Algebras, Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. | MR 1656031 | Zbl 0913.46054

[12] Blanchard E., Déformations de C * -algèbres de Hopf, Bull. Soc. Math. France 124 (1996) 141-215. | Numdam | MR 1395009 | Zbl 0851.46040

[13] Bost J.-B., Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math. 101 (1990) 261-333. | MR 1062964 | Zbl 0719.46038

[14] Brylinski J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, vol. 107, Birkhäuser, Basel, 1993. | MR 1197353 | Zbl 0823.55002

[15] Bouwknegt P., Mathai V., D-branes, B-fields and twisted K-theory, J. High Energy Phys. 3 (2000) 7-11. | MR 1756434 | Zbl 0959.81037

[16] Bouwknegt P., Carey A., Mathai V., Murray M., Stevenson D., Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys. 228 (2002) 17-45. | MR 1911247 | Zbl 1036.19005

[17] Connes A., Cyclic Cohomology and the Transverse Fundamental Class of a Foliation, in: Pitman Research Notes Math. Ser., vol. 123, Longman Sci., Harlow, 1986, pp. 52-144. | MR 866491 | Zbl 0647.46054

[18] Connes A., Skandalis G., The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. Kyoto Univ. 20 (1984) 1139-1183. | MR 775126 | Zbl 0575.58030

[19] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003) 681-721. | MR 2016690 | Zbl 1041.58007

[20] Crainic M., Moerdijk I., A homology theory for étale groupoids, J. Reine Angew. Math. 521 (2000) 25-46. | MR 1752294 | Zbl 0954.22002

[21] Dixmier J., C * -Algebras, North Holland, Amsterdam, 1977. | MR 458185 | Zbl 0372.46058

[22] Dixmier J., Douady A., Champs continus d’espaces hilbertiens et de C * -algèbres, Bull. Soc. Math. France 91 (1963) 227-284. | Numdam | MR 163182 | Zbl 0127.33102

[23] Donovan P., Karoubi M., Graded Brauer groups and K-theory with local coefficients, Inst. Hautes Études Sci. Publ. 38 (1970) 5-25. | Numdam | MR 282363 | Zbl 0207.22003

[24] Dupont J., Curvature and Characteristic Classes, Lecture Notes in Mathematics, vol. 640, Springer, Berlin, 1978. | MR 500997 | Zbl 0373.57009

[25] Dupré M.J., Gillette R.M., Banach Bundles, Banach Modules and Automorphisms of C * -Algebras, Pitman Research Notes in Mathematics, vol. 92, 1983. | MR 721812 | Zbl 0536.46048

[26] Fell J., Doran R., Representations of C * -Algebras, Locally Compact Groups, and Banach ∗-Algebraic Bundles, vol. 2, Pure and Applied Mathematics, vol. 126, Academic Press, Boston, MA, 1988. | Zbl 0652.46051

[27] Freed D., The Verlinde algebra is twisted equivariant K-theory, Turkish J. Math. 25 (2001) 159-167. | MR 1829086 | Zbl 0971.55006

[28] Fulman I., Muhly P., Bimodules, spectra, and Fell bundles, Israel J. Math. 108 (1998) 193-215. | MR 1669380 | Zbl 0917.46064

[29] Gabriel P., Zisman M., Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer, Berlin, 1967. | MR 210125 | Zbl 0186.56802

[30] Giraud J., Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer, Berlin, 1971. | MR 344253 | Zbl 0226.14011

[31] Grothendieck A., Le groupe de Brauer, Séminaire Bourbaki (1964/65), Collection Hors Série de la S.M.F. 9 (1995) 199-219. | Numdam | MR 1608798 | Zbl 0186.54702

[32] Guillemin V., Sternberg S., Supersymmetry and Equivariant de Rham Theory, Springer, Berlin, 1999. | MR 1689252 | Zbl 0934.55007

[33] Haefliger A., Groupoïdes d'holonomie et classifiants, Astérisque 116 (1984) 70-97. | MR 755163 | Zbl 0562.57012

[34] Higson N., On a technical theorem of Kasparov, J. Funct. Anal. 73 (1987) 107-112. | MR 890657 | Zbl 0623.46035

[35] Higson N., The Baum-Connes conjecture, in: Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998), Doc. Math., Extra vol. II, 1998, pp. 637-646, (electronic). | MR 1648112 | Zbl 0911.46041

[36] Higson N., Lafforgue V., Skandalis G., Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2) (2002) 330-354. | MR 1911663 | Zbl 1014.46043

[37] Higson N., Roe J., Yu G., A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114 (1993) 85-97. | MR 1219916 | Zbl 0792.55001

[38] Hilsum M., Skandalis G., Morphismes K-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes), Ann. Sci. Éc. Norm. Sup. 20 (1987) 325-390. | Numdam | MR 925720 | Zbl 0656.57015

[39] Hitchin N., Lectures on special Lagrangian submanifolds, AMS/IP Stud. Adv. Math. 23 (1999) 151-182. | MR 1876068 | Zbl 1079.14522

[40] Kellendonk J., Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys. 7 (7) (1995) 1133-1180. | MR 1359991 | Zbl 0847.52022

[41] Koosis P., An irreducible unitary representation of a compact group is finite dimensional, Proc. Amer. Math. Soc. 8 (1957) 712-715. | MR 87888 | Zbl 0079.32802

[42] Kumjian A., Fell Bundles over groupoids, Proc. Amer. Math. Soc. 128 (1998) 1115-1125. | MR 1443836 | Zbl 0891.46038

[43] Kumjian A., Muhly P., Renault J., Williams D., The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998) 901-954. | MR 1646047 | Zbl 0916.46050

[44] Le Gall P.-Y., Théorie de Kasparov équivariante et groupoïdes, 16 (1999) 361-390. | MR 1686846 | Zbl 0932.19004

[45] Lupercio E., Uribe B., Gerbes over orbifolds and twisted K-theory, Comm. Math. Phys. 245 (2004) 449-489. | MR 2045679 | Zbl 1068.53034

[46] Mathai V., Stevenson D., Chern character in twisted K-theory, equivariant and holomorphic cases, Comm. Math. Phys. 236 (2003) 161-186. | MR 1977885 | Zbl 1030.19004

[47] Meinrenken E., The basic gerbe over a compact simple Lie group, Enseign. Math. (2) 49 (2003) 307-333. | MR 2026898 | Zbl 1061.53034

[48] Mickelsson J., Gerbes, (twisted) K-theory, and the supersymmetric WZW model, hep-th/0206139.

[49] Minasian R., Moore G., K-theory and Ramond-Ramond charge, J. High Energy Phys. 11 (1997) 2-7. | MR 1606278 | Zbl 0949.81511

[50] Monthubert B., Groupoids of manifolds with corners and index theory, in: Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 147-157. | MR 1855248 | Zbl 1043.35148

[51] Mrčun J., Functoriality of the bimodule associated to a Hilsum-Skandalis map, 18 (1999) 235-253. | MR 1722796 | Zbl 0938.22002

[52] Moerdijk I., Orbifolds as groupoids: an introduction, in: Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., vol. 310, 2002, pp. 205-222. | MR 1950948 | Zbl 1041.58009

[53] Moerdijk I., Introduction to the language of stacks and gerbes, math.AT/0212266.

[54] Moerdijk I., Pronk D.A., Orbifolds, sheaves and groupoids, 12 (1997) 3-21. | MR 1466622 | Zbl 0883.22005

[55] Muhly P., Bundles over groupoids, in: Groupoids in Analysis, Geometry and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2000, pp. 67-82. | MR 1855243 | Zbl 1011.46052

[56] Muhly P., Renault J., Williams D., Equivalence and isomorphism for groupoid C * -algebras, J. Operator Theory 17 (1987) 3-22. | MR 873460 | Zbl 0645.46040

[57] Paterson A., The analytic index for proper, Lie groupoid actions, in: Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 115-135. | MR 1855246 | Zbl 0995.19005

[58] Pedersen G., C * -Algebras and their Automorphism Groups, London Mathematical Society Monographs, vol. 14, Academic Press, London, 1979. | MR 548006 | Zbl 0416.46043

[59] Pressley A., Segal G., Loop Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1986. | MR 900587 | Zbl 0618.22011

[60] Raeburn I., Taylor J., Continuous trace C * -algebras with given Dixmier-Douady class, J. Austral. Math. Soc. Ser. A 38 (1985) 394-407. | MR 779202 | Zbl 0627.46068

[61] Renault J., A Groupoid Approach to C * -Algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. | MR 584266 | Zbl 0433.46049

[62] Renault J., Représentation des produits croisés d'algèbres de groupoïdes, J. Operator Theory 18 (1987) 67-97. | MR 912813 | Zbl 0659.46058

[63] Rosenberg J., Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. Soc. Ser. A 47 (1989) 368-381. | MR 1018964 | Zbl 0695.46031

[64] Schweitzer L.B., A short proof that M n A is local if A is local and Fréchet, Internat. J. Math. 3 (4) (1992) 581-589. | MR 1168361 | Zbl 0804.46054

[65] Segal G., Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 129-151. | Numdam | MR 234452 | Zbl 0199.26202

[66] Segal G., Fredholm complexes, Quart. J. Math. Oxford Ser. (2) 21 (1970) 385-402. | MR 271930 | Zbl 0213.25403

[67] Skandalis G., Tu J.L., Yu G., The coarse Baum-Connes conjecture and groupoids, Topology 41 (4) (2002) 807-834. | MR 1905840 | Zbl 1033.19003

[68] Tu J.-L., La conjecture de Novikov pour les feuilletages hyperboliques, 16 (1999) 129-184. | MR 1671260 | Zbl 0932.19005

[69] Tu J.-L., La conjecture de Baum-Connes pour les feuilletages moyennables, 17 (3) (1999) 215-264. | MR 1703305 | Zbl 0939.19001

[70] Tuynman G.M., Wiegerinck G.M., Central extensions and physics, J. Geom. Phys. 4 (1987) 207-258. | MR 948561 | Zbl 0649.58014

[71] Yamagami S., On primitive ideal spaces of C * -algebras over certain locally compact groupoids, in: Mappings of Operator Algebras (Philadelphia, PA, 1988), Progress in Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 199-204. | MR 1103378 | Zbl 0733.46035

[72] Wegge-Olsen N.E., K-theory and C * -algebras, A friendly approach, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. | MR 1222415 | Zbl 0780.46038

[73] Weil A., Sur les théorèmes de De Rham, Comment. Math. Helv. 26 (1959) 119-145. | MR 50280 | Zbl 0047.16702

[74] Weinstein A., Xu P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991) 159-189. | MR 1103911 | Zbl 0722.58021

[75] Witten E., D-branes and K-theory, J. High Energy Phys. 12 (1998) 19-44. | MR 1674715 | Zbl 0959.81070

[76] Witten E., Overview of K-theory applied to strings, Internat. J. Modern Phys. A 16 (2001) 693-706. | MR 1827946 | Zbl 0980.81049

[77] Xu P., Morita equivalent symplectic groupoids, Math. Sci. Res. Inst. Publ. 20 (1991) 291-311. | MR 1104935 | Zbl 0733.58013