[116] Equivariant Characteristic Classes in the Cartan Model

  • Raoul Bott
  • Loring W. Tu
Part of the Contemporary Mathematicians book series (CM)


There is also a differential geometric definition of equivariant characteristic classes in terms of the curvature of a connection on P (3)(4). However, there does not seem to be an explanation or justification in the literature bridging the two approaches. The modest purpose of this note is to show the compatibility of the usual differential geometric formulation of equivariant characteristic classes with the topological one. We have also tried to be as self-contained as possible, which partly explains the length of this article.


  1. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), pp. 1-28.MathSciNetCrossRefGoogle Scholar
  2. M. Audin, The Topology of Torus Actions on Symplectic Manifolds (Birkhäuser, Basel, 1991).CrossRefGoogle Scholar
  3. N. Berline and M. Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sc. Paris, Série I, t. 295 (1982), pp. 539–540.Google Scholar
  4. H. Cartan, Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés oú opére un groupe de Lie, in Colloque de Topologie (espaces fibré) Bruxelles 1950 (Centre Belge de Recherches Mathématiques, Louvain, Belgium), pp. 15–27.Google Scholar
  5. H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in Colloque de Topologie (espaces fibré) Bruxelles 1950 (Centre Belge de Recherches Mathématiques, Louvain, Belgium), pp. 57–71.Google Scholar
  6. V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
  7. A. Hatcher, Algebraic Topology I, to be published by Cambridge University Press, also available online at
  8. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1 and 2 (Wiley, New York, 1963 and 1969).Google Scholar
  9. V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), pp. 85-110.MathSciNetCrossRefGoogle Scholar
  10. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 2, second edition(Publish or Perish, Berkeley, 1979).Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

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