Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Groupoid cohomology and extensions


Author: Jean-Louis Tu
Journal: Trans. Amer. Math. Soc. 358 (2006), 4721-4747
MSC (2000): Primary 22A22, 20J06; Secondary 55Nxx
DOI: https://doi.org/10.1090/S0002-9947-06-03982-1
Published electronically: May 9, 2006
MathSciNet review: 2231869
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.


References [Enhancements On Off] (What's this?)

  • 1. Artin, M. et al., Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique, Lecture Notes in Mathematics 269, 270, 305 (Springer, 1972-1973). MR 0354652 (50:7130), MR 0354653 (50:7131), MR 0354654 (50:7132)
  • 2. Atiyah, M. F.; Wall, C. T. C., Cohomology of groups. 1967. Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 94-115 Thompson, Washington, D.C. MR 0219512 (36:2593)
  • 3. Crainic, M. and Moerdijk, I. A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25-46. MR 1752294 (2001f:58039)
  • 4. Deligne, P., Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. No. 44 (1974), 5-77. MR 0498552 (58:16653b)
  • 5. Godement, R., Topologie algébrique et théorie des faisceaux. Hermann, Paris. Third Edition (1973). MR 0345092 (49:9831)
  • 6. Haefliger, A., Differential cohomology. Differential topology (Varenna, 1976), pp. 19-70, Liguori, Naples, 1979. MR 0660658 (83k:17014)
  • 7. Haefliger, A., Groupoïdes d'holonomie et classifiants, Structure transverse des feuilletages, Toulouse 1982, Astérisque 116 (1984), 70-97. MR 0755163 (86c:57026a)
  • 8. Hilsum, M., and Skandalis, G., Morphismes $ K$-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes). Ann. Sci. École Norm. Sup. 20 (1987), 325-390. MR 0925720 (90a:58169)
  • 9. Holt, D. F., An interpretation of the cohomology groups $ H\sp{n}(G,\,M)$. J. Algebra 60 (1979), no. 2, 307-320. MR 0549932 (80m:20040)
  • 10. Kumjian, A., On equivariant sheaf cohomology and elementary $ C\sp *$-bundles. J. Operator Theory 20 (1988), no. 2, 207-240. MR 1004121 (90h:46102)
  • 11. Kumjian, A., Muhly, P., Renault, J. and Williams, D., The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998), 901-954. MR 1646047 (2000b:46122)
  • 12. Landsman, N. P., Quantized reduction as a tensor product. Quantization of singular symplectic quotients, 137-180, Progr. Math., 198, Birkhäuser, Basel, 2001. MR 1938555 (2004c:53138)
  • 13. Le Gall, P.-Y., Théorie de Kasparov équivariante et groupoïdes, $ K$-Theory 16 (1999), 361-390.MR 1686846 (2000f:19006)
  • 14. MacLane, S., Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971.MR 0354798 (50:7275)
  • 15. Moerdijk, I., Classifying toposes and foliations. Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 189-209. MR 1112197 (92i:57028)
  • 16. Moerdijk, I., Proof of a conjecture of A. Haefliger. Topology 37 (1998), no. 4, 735-741. MR 1607724 (99b:18015)
  • 17. Moerdijk, I., Lie groupoids, gerbes, and non-abelian cohomology. $ K$-Theory 28 (2003), no. 3, 207-258. MR 2017529 (2005b:58024)
  • 18. Moore, C., Extensions and low dimensional cohomology theory of locally compact groups. I, II. Trans. Amer. Math. Soc. 113 (1964), 40-63; ibid., 64-86. MR 0171880 (30:2106)
  • 19. Moore, C., Group extensions and cohomology for locally compact groups. III, IV. Trans. Amer. Math. Soc. 221, no. 1 (1976), 1-33; ibid., 35-58. MR 0414775 (54:2867), MR 0414776 (54:2868)
  • 20. Mrcun, J., Functoriality of the bimodule associated to a Hilsum-Skandalis map, $ K$-Theory 18 (1999), 235-253. MR 1722796 (2001k:22004)
  • 21. Pronk, Dorette A., Etendues and stacks as bicategories of fractions. Compositio Math. 102 (1996), no. 3, 243-303. MR 1401424 (97d:18011)
  • 22. Renault, J., A groupoid approach to $ C\sp{*} $-algebras. Lecture Notes in Mathematics, 793, Springer, Berlin, 1980. MR 0584266 (82h:46075)
  • 23. Segal, G., Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math. no. 34, (1968), pp 105-112. MR 0232393 (38:718)
  • 24. Tu, J.L., Xu, P. and Laurent-Gengoux, C., Twisted $ K$-theory of differentiable stacks. Ann. Sci. École Norm. Sup. (4) 37 (2004), 841-910. MR 2119241 (2005k:58037)
  • 25. Wigner, D., Algebraic cohomology of topological groups. Trans. Amer. Math. Soc. 178 (1973), 83-93. MR 0338132 (49:2898)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22A22, 20J06, 55Nxx

Retrieve articles in all journals with MSC (2000): 22A22, 20J06, 55Nxx


Additional Information

Jean-Louis Tu
Affiliation: Institut de Mathématiques, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France
Address at time of publication: Université Paul Verlaine-Metz, LMAM-CNRS UMR 7122, Ile de Saulcy, 57000 Metz, France
Email: tu@univ-metz.fr

DOI: https://doi.org/10.1090/S0002-9947-06-03982-1
Keywords: Groupoid, sheaf cohomology, Haefliger's cohomology, Brauer group of a groupoid
Received by editor(s): June 28, 2004
Published electronically: May 9, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society