@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}, volume = {37}, year = {2004}, pages = {841-910}, doi = {10.1016/j.ansens.2004.10.002}, zbl = {1069.19006}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 37 (2004) pp. 841-910. doi : 10.1016/j.ansens.2004.10.002. http://gdmltest.u-ga.fr/item/ASENS_2004_4_37_6_841_0/
[1] Twisted orbifold K-theory, Comm. Math. Phys. 237 (2003) 533-556. | MR 1993337 | Zbl 1051.57022
, ,[2] 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.
et al. ,[3] K-theory. Lecture notes by D.W. Anderson, 1967. | MR 224083
,[4] K-theory past and present, math.KT/0012213. | MR 2091892
,[5] Twisted K-theory, math.KT/0407054.
, ,[6] Chern character for discrete groups, in: A fête of topology, Academic Press, New York, 1988, pp. 163-232. | MR 928402 | Zbl 0656.55005
, ,[7] Classifying space for proper actions and K-theory of group -algebras, in: -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] -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] K-Theory for Operator Algebras, Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. | MR 1656031 | Zbl 0913.46054
,[12] Déformations de -algèbres de Hopf, Bull. Soc. Math. France 124 (1996) 141-215. | Numdam | MR 1395009 | Zbl 0851.46040
,[13] Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math. 101 (1990) 261-333. | MR 1062964 | Zbl 0719.46038
,[14] Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, vol. 107, Birkhäuser, Basel, 1993. | MR 1197353 | Zbl 0823.55002
,[15] D-branes, B-fields and twisted K-theory, J. High Energy Phys. 3 (2000) 7-11. | MR 1756434 | Zbl 0959.81037
, ,[16] Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys. 228 (2002) 17-45. | MR 1911247 | Zbl 1036.19005
, , , , ,[17] 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] The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. Kyoto Univ. 20 (1984) 1139-1183. | MR 775126 | Zbl 0575.58030
, ,[19] Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003) 681-721. | MR 2016690 | Zbl 1041.58007
,[20] A homology theory for étale groupoids, J. Reine Angew. Math. 521 (2000) 25-46. | MR 1752294 | Zbl 0954.22002
, ,[21] -Algebras, North Holland, Amsterdam, 1977. | MR 458185 | Zbl 0372.46058
,[22] Champs continus d’espaces hilbertiens et de -algèbres, Bull. Soc. Math. France 91 (1963) 227-284. | Numdam | MR 163182 | Zbl 0127.33102
, ,[23] 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] Curvature and Characteristic Classes, Lecture Notes in Mathematics, vol. 640, Springer, Berlin, 1978. | MR 500997 | Zbl 0373.57009
,[25] Banach Bundles, Banach Modules and Automorphisms of -Algebras, Pitman Research Notes in Mathematics, vol. 92, 1983. | MR 721812 | Zbl 0536.46048
, ,[26] Representations of -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] The Verlinde algebra is twisted equivariant K-theory, Turkish J. Math. 25 (2001) 159-167. | MR 1829086 | Zbl 0971.55006
,[28] Bimodules, spectra, and Fell bundles, Israel J. Math. 108 (1998) 193-215. | MR 1669380 | Zbl 0917.46064
, ,[29] Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer, Berlin, 1967. | MR 210125 | Zbl 0186.56802
, ,[30] Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer, Berlin, 1971. | MR 344253 | Zbl 0226.14011
,[31] 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] Supersymmetry and Equivariant de Rham Theory, Springer, Berlin, 1999. | MR 1689252 | Zbl 0934.55007
, ,[33] Groupoïdes d'holonomie et classifiants, Astérisque 116 (1984) 70-97. | MR 755163 | Zbl 0562.57012
,[34] On a technical theorem of Kasparov, J. Funct. Anal. 73 (1987) 107-112. | MR 890657 | Zbl 0623.46035
,[35] 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] Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2) (2002) 330-354. | MR 1911663 | Zbl 1014.46043
, , ,[37] A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114 (1993) 85-97. | MR 1219916 | Zbl 0792.55001
, , ,[38] 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] Lectures on special Lagrangian submanifolds, AMS/IP Stud. Adv. Math. 23 (1999) 151-182. | MR 1876068 | Zbl 1079.14522
,[40] Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys. 7 (7) (1995) 1133-1180. | MR 1359991 | Zbl 0847.52022
,[41] 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] Fell Bundles over groupoids, Proc. Amer. Math. Soc. 128 (1998) 1115-1125. | MR 1443836 | Zbl 0891.46038
,[43] The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998) 901-954. | MR 1646047 | Zbl 0916.46050
, , , ,[44] Théorie de Kasparov équivariante et groupoïdes, 16 (1999) 361-390. | MR 1686846 | Zbl 0932.19004
,[45] Gerbes over orbifolds and twisted K-theory, Comm. Math. Phys. 245 (2004) 449-489. | MR 2045679 | Zbl 1068.53034
, ,[46] Chern character in twisted K-theory, equivariant and holomorphic cases, Comm. Math. Phys. 236 (2003) 161-186. | MR 1977885 | Zbl 1030.19004
, ,[47] The basic gerbe over a compact simple Lie group, Enseign. Math. (2) 49 (2003) 307-333. | MR 2026898 | Zbl 1061.53034
,[48] Gerbes, (twisted) K-theory, and the supersymmetric WZW model, hep-th/0206139.
,[49] K-theory and Ramond-Ramond charge, J. High Energy Phys. 11 (1997) 2-7. | MR 1606278 | Zbl 0949.81511
, ,[50] 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] Functoriality of the bimodule associated to a Hilsum-Skandalis map, 18 (1999) 235-253. | MR 1722796 | Zbl 0938.22002
,[52] 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] Introduction to the language of stacks and gerbes, math.AT/0212266.
,[54] Orbifolds, sheaves and groupoids, 12 (1997) 3-21. | MR 1466622 | Zbl 0883.22005
, ,[55] 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] Equivalence and isomorphism for groupoid -algebras, J. Operator Theory 17 (1987) 3-22. | MR 873460 | Zbl 0645.46040
, , ,[57] 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] -Algebras and their Automorphism Groups, London Mathematical Society Monographs, vol. 14, Academic Press, London, 1979. | MR 548006 | Zbl 0416.46043
,[59] Loop Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1986. | MR 900587 | Zbl 0618.22011
, ,[60] Continuous trace -algebras with given Dixmier-Douady class, J. Austral. Math. Soc. Ser. A 38 (1985) 394-407. | MR 779202 | Zbl 0627.46068
, ,[61] A Groupoid Approach to -Algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. | MR 584266 | Zbl 0433.46049
,[62] Représentation des produits croisés d'algèbres de groupoïdes, J. Operator Theory 18 (1987) 67-97. | MR 912813 | Zbl 0659.46058
,[63] 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] A short proof that is local if A is local and Fréchet, Internat. J. Math. 3 (4) (1992) 581-589. | MR 1168361 | Zbl 0804.46054
,[65] Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 129-151. | Numdam | MR 234452 | Zbl 0199.26202
,[66] Fredholm complexes, Quart. J. Math. Oxford Ser. (2) 21 (1970) 385-402. | MR 271930 | Zbl 0213.25403
,[67] The coarse Baum-Connes conjecture and groupoids, Topology 41 (4) (2002) 807-834. | MR 1905840 | Zbl 1033.19003
, , ,[68] La conjecture de Novikov pour les feuilletages hyperboliques, 16 (1999) 129-184. | MR 1671260 | Zbl 0932.19005
,[69] La conjecture de Baum-Connes pour les feuilletages moyennables, 17 (3) (1999) 215-264. | MR 1703305 | Zbl 0939.19001
,[70] Central extensions and physics, J. Geom. Phys. 4 (1987) 207-258. | MR 948561 | Zbl 0649.58014
, ,[71] On primitive ideal spaces of -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] K-theory and -algebras, A friendly approach, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. | MR 1222415 | Zbl 0780.46038
,[73] Sur les théorèmes de De Rham, Comment. Math. Helv. 26 (1959) 119-145. | MR 50280 | Zbl 0047.16702
,[74] Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991) 159-189. | MR 1103911 | Zbl 0722.58021
, ,[75] D-branes and K-theory, J. High Energy Phys. 12 (1998) 19-44. | MR 1674715 | Zbl 0959.81070
,[76] Overview of K-theory applied to strings, Internat. J. Modern Phys. A 16 (2001) 693-706. | MR 1827946 | Zbl 0980.81049
,[77] Morita equivalent symplectic groupoids, Math. Sci. Res. Inst. Publ. 20 (1991) 291-311. | MR 1104935 | Zbl 0733.58013
,