On the Weak Homotopy Type of Étale Groupoids

  • Ieke Moerdijk
Conference paper
Part of the Progress in Mathematics book series (PM, volume 145)


Etale groupoids play a central role in the theory of foliations. Well-known examples include the Haefliger groupoid Γ q which classifies C -foliations of codimension q [H71] and the holonomy groupoid of any foliation [W83]. In particular, invariants of leaf spaces of foliations are usually defined in terms of the classifying space or the C*-algebra associated to this holonomy groupoid (see [C, H84, Mo, BN] and many others).


Geometric Realization Riemannian Foliation Topological Category Local Homeomorphism Sheaf Cohomology 
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  1. [B]
    R. Bott, Characteristic classes and foliations, in Springer LNM 279 (1972), 1–94.MathSciNetGoogle Scholar
  2. [BK]
    A.K. Bousfield, D.M. Kan, Homotopy Limits, Completions and Localizations, Springer LNM 304, 1972.Google Scholar
  3. [BN]
    J.-L. Brylinski, V. Nistor, Cyclic cohomology of étale groupoids, K-Theory 8 (1994), 341–365.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [C]
    A. Connes, Non Commutative Geometry, Academic Press, 1994.Google Scholar
  5. [vE]
    W.T. van Est, Rapport sur les S-atlas, Astérisque 116 (1984), 235–292.Google Scholar
  6. [H71]
    A. Haefliger, Homotopy and integrability, in: Springer LNM 197(1971), 133–163.MathSciNetGoogle Scholar
  7. [H76]
    A. Haefliger, Differentiate Cohomology, CIME, Varenna, 1976.Google Scholar
  8. [H84]
    A. Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97.MathSciNetGoogle Scholar
  9. [H92]
    A. Haefliger, Cohomology theory for étale topological groupoids, unpublished manuscript, 1992.Google Scholar
  10. [CWM]
    S. Mac Lane, Categories for The Working Mathematician, Springer-Verlag, 1971.zbMATHGoogle Scholar
  11. [M91]
    I. Moerdijk, Classifying spaces and foliations, Ann. Inst. Fourier 41 (1991), 189–209.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [M95]
    I. Moerdijk, Classifying Spaces and Classifying Topoi, Springer LNM 1616 (1995).Google Scholar
  13. [MP]
    I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory (to appear).Google Scholar
  14. [Mo]
    P. Molino, Riemannian Foliations, Birkhäuser, 1988.zbMATHGoogle Scholar
  15. [S68]
    G.B. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES 34 (1968), 105–112.zbMATHGoogle Scholar
  16. [S74]
    G.B. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.Google Scholar
  17. [S78]
    G.B. Segal, Classifying spaces related to foliations, Topology 17 (1978), 367–382.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [SGA4]
    M. Artin, A. Grothendieck, J.-L. Verdier, Théorie de topos et cohomologie étale des schémas, tome 2, Springer LNM 270, (1972).Google Scholar
  19. [W83]
    H. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), 51–75.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Ieke Moerdijk
    • 1
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands

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