São Paulo Journal of Mathematical Sciences

, Volume 13, Issue 2, pp 539–596 | Cite as

Equivariant de Rham cohomology: theory and applications

  • Oliver GoertschesEmail author
  • Leopold Zoller


This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.


Lie group actions Equivariant Cohomology Cartan model Equivariant formality Fixed points 



Parts of this paper stem from the first named author’s lectures at the University of Hamburg in 2012, and at the Philipps University of Marburg in 2018. We would like to thank the participants of these courses for their interest in the topic and their valuable comments. We are grateful to Michèle Vergne for several remarks on a previous version of this paper. We are especially indebted to Jeffrey Carlson for several enlightening discussions, as well as for a very thorough reading of a previous version and numerous suggestions that improved the presentation of this paper. The second named author is supported by the German Academic Scholarship foundation.


  1. 1.
    Allday, C.: A family of unusual torus group actions. Group Actions on Manifolds (Boulder, Colo., 1983), vol. 36, pp. 107–111, Contemp. Math. American Mathematical Society, Providence (1985)Google Scholar
  2. 2.
    Allday, C., Franz, M., Puppe, V.: Equivariant cohomology, syzygies and orbit structure. Trans. Am. Math. Soc. 366, 6567–6589 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar
  4. 4.
    Álvarez López, J.A., Kordykov, Y.A.: Lefschetz distribution of Lie foliations. In: \(C^*\)-algebras and elliptic theory II, pp. 1–40. Trends in Mathematics, Basel (2008)Google Scholar
  5. 5.
    Atiyah, M.F.: Elliptic Operators and Compact Groups. Lecture Notes in Mathematics, vol. 401. Springer, Berlin (1974)zbMATHCrossRefGoogle Scholar
  6. 6.
    Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14(1), 1–15 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308(1505), 523–615 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969)zbMATHGoogle Scholar
  9. 9.
    Audin, M.: Torus Actions on Symplectic Manifolds, 2nd revised edn. Progress in Mathematics, vol. 93. Birkhäuser Verlag, Basel (2004)Google Scholar
  10. 10.
    Bazzoni, G., Goertsches, O.: \(K\)-cosymplectic manifolds. Ann. Glob. Anal. Geom. 47(3), 239–270 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer, Berlin (2004)Google Scholar
  12. 12.
    Borel, A.: Seminar on Transformation Groups. With contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, no. 46. Princeton University Press, Princeton (1960)Google Scholar
  13. 13.
    Bott, R.: An introduction to equivariant cohomology. Quantum field theory: perspective and prospective (Les Houches, 1998), vol. 530, pp. 35–56, NATO Sci. Ser. C Math. Phys. Sci. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  14. 14.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)zbMATHCrossRefGoogle Scholar
  15. 15.
    Bourbaki, N.: Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées. Actualités Scientifiques et Industrielles, no. 1364. Hermann, Paris (1975)Google Scholar
  16. 16.
    Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972)zbMATHGoogle Scholar
  17. 17.
    Bredon, G.E.: The free part of a torus action and related numerical equalities. Duke Math. J. 41, 843–854 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Carlson, J.D.: Equivariant formality of isotropic torus actions. J. Homotopy Relat. Struct. 14(1), 199–234 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Carlson, J.D., Fok, C.-K.: Equivariant formality of isotropy actions. J. Lond. Math. Soc. 97(3), 470–494 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Carlson, J.D., Goertsches, O., He, C., Mare, A.-L.: The equivariant cohomology ring of a cohomogeneity-one action. Preprint. arXiv:1802.02304
  21. 21.
    Cartan, É.: Sur les invariants intégraux de certains espaces homogènes clos et les propriétés topologiques de ces espaces. Ann. Soc. Pol. Math. 8, 181–225 (1929)zbMATHGoogle Scholar
  22. 22.
    Cartan, H.: Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie. Colloque de topologie C.B.R.M., Bruxelles, pp. 15–27 (1950)Google Scholar
  23. 23.
    Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal. Colloque de topologie, C.B.R.M., Bruxelles, pp. 57–71 (1950)Google Scholar
  24. 24.
    Chang, T., Skjelbred, T.: The topological Schur lemma and related results. Ann. Math. 2(100), 307–321 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Chevalley, C.: The Betti numbers of the exceptional simple Lie groups. In: Proceedings of the International Congress of Mathematicians, Cambridge, pp. 21–24, 1950. American Mathematical Society, Providence, (1952)Google Scholar
  26. 26.
    Duflo, M., Vergne, M.: Cohomologie équivariante et descente. Sur la cohomologie équivariante des variétés différentiables. Astérisque No. 215, pp. 5–108 (1993)Google Scholar
  27. 27.
    Duflo, M., Vergne, M.: Orbites coadjointes et cohomologie équivariante. The orbit method in representation theory (Copenhagen, 1988), pp. 11–60, Prog. Math., vol. 82. Birkhäuser Boston (1990)Google Scholar
  28. 28.
    Duflot, J.: Smooth toral actions. Topology 22(3), 253–265 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Félix, Y., Oprea, J., Tanré, D.: Algebraic Models in Geometry. Oxford Graduate Texts in Mathematics, vol. 17. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  30. 30.
    Fok, C.-K.: Cohomology and \(K\)-theory of compact Lie groups (2018). Preprint. Accessed 5 Oct 2018
  31. 31.
    Franz, M.: Big polygon spaces. Int. Math. Res. Not. IMRN 24, 13379–13405 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Franz, M., Puppe, V.: Freeness of equivariant cohomology and mutants of compactified representations. In: Harada, M., et al. (eds.) Toric Topology (Osaka, 2006), Contemporary Mathematics, vol. 460 (2008)Google Scholar
  33. 33.
    Franz, M., Puppe, V.: Exact cohomology sequences with integral coefficients for torus actions. Transform. Groups 12(1), 65–76 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Franz, M., Puppe, V.: Exact sequences for equivariantly formal spaces. C. R. Math. Acad. Sci. Soc. R. Can. 33, 1–10 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ginzburg, V.: Equivariant cohomology and Kähler geometry. Funct. Anal. Appl. 21(4), 271–283 (1987). (in Russian) zbMATHCrossRefGoogle Scholar
  36. 36.
    Goertsches, O.: The equivariant cohomology of isotropy actions on symmetric spaces. Doc. Math. 17, 79–94 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Goertsches, O., Hagh Shenas Noshari, S.: Equivariant formality of isotropy actions on homogeneous spaces defined by Lie group automorphisms. J. Pure Appl. Algebra 220, 2017–2028 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Goertsches, O., Hagh Shenas Noshari, S., Mare, A.-L.: On the equivariant cohomology of hyperpolar actions on symmetric spaces. Preprint. arXiv:1808.10630
  39. 39.
    Goertsches, O., Mare, A.-L.: Equivariant cohomology of cohomogeneity one actions. Topol. Appl. 167, 36–52 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Goertsches, O., Mare, A.-L.: Non-abelian GKM theory. Math. Z. 277(1–2), 1–27 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Goertsches, O., Nozawa, H., Töben, D.: Equivariant cohomology of K-contact manifolds. Math. Ann. 354(4), 1555–1582 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Goertsches, O., Rollenske, S.: Torsion in equivariant cohomology and Cohen-Macaulay \(G\)-actions. Transform. Groups 16(4), 1063–1080 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Goertsches, O., Töben, D.: Torus actions whose cohomology is Cohen-Macaulay. J. Topol. 3(4), 819–846 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Goertsches, O., Töben, D.: Equivariant basic cohomology of Riemannian foliations. J. Reine Angew. Math. 745, 1–40 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Goertsches, O., Wiemeler, M.: Positively curved GKM-manifolds. Int. Math. Res. Not. IMRN 22, 12015–12041 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology: Cohomology of Principal Bundles and Homogeneous Spaces, Connections, Curvature, and Cohomology, vol. III. Academic Press, Cambridge (1976)zbMATHGoogle Scholar
  48. 48.
    Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra 91(1–3), 137–142 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Guillemin, V., Ginzburg, V., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions. Mathematical Surveys and Monographs, vol. 98. AMS, Providence (2002)zbMATHCrossRefGoogle Scholar
  50. 50.
    Guillemin, V., Holm, T.S.: GKM theory for torus actions with non-isolated fixed points. Int. Math. Res. Not. IMRN 40, 2105–2124 (2004)zbMATHCrossRefGoogle Scholar
  51. 51.
    Guillemin, V., Holm, T., Zara, C.: A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebraic Combin. 23(1), 21–41 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
  53. 53.
    Guillemin, V., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Mathematics Past and Present. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
  54. 54.
    Guillemin, V., Zara, C.: Equivariant de Rham theory and graphs. Asian J. Math 3(1), 49–76 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Hagh Shenas Noshari, S.: On the equivariant cohomology of isotropy actions. Dissertation, Philipps University of Marburg (2018)Google Scholar
  56. 56.
    Harada, M., Henriques, A., Holm, T.S.: Computation of generalized equivariant cohomologies of KacMoody flag varieties. Adv. Math. 197(1), 198–221 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    He, C.: Localization of certain odd-dimensional manifolds with torus actions. Preprint. arXiv:1608.04392
  58. 58.
    Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1975)CrossRefGoogle Scholar
  59. 59.
    Kane, R.M.: Reflection groups and invariant theory. In: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 5. Springer, New York (2001)Google Scholar
  60. 60.
    Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press, Princeton (1984)zbMATHGoogle Scholar
  61. 61.
    Knapp, A.W.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser Boston, Inc., Boston (2002)zbMATHGoogle Scholar
  62. 62.
    Kobayashi, S.: Fixed points of isometries. Nagoya Math. J. 13, 63–68 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Kumar, S.: Kac-Moody Groups, Their Flag Varieties, and Representation Theory. Birkhäuser, Basel (2001)Google Scholar
  64. 64.
    Mare, A.-L.: An introduction to flag manifolds, Notes for the Summer School on Combinatorial Models in Geometry and Topology of Flag Manifolds, Regina (2007). Accessed 5 Oct 2018
  65. 65.
    McCleary, J.: A User’s Guide to Spectral Sequences. Cambridge University Press, Cambridge (2000)zbMATHCrossRefGoogle Scholar
  66. 66.
    Meinrenken, E.: Equivariant Cohomology and the Cartan Model, Encyclopedia of Mathematical Physics. Elsevier, Amsterdam (2006)Google Scholar
  67. 67.
    Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  68. 68.
    Molino, P.: Riemannian foliations, with appendices by G. Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. Birkhäuser Boston Inc., Boston (1988)Google Scholar
  69. 69.
    Nicolaescu, L.: On a theorem of Henri Cartan concerning the equivariant cohomology. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 45(1), 17–38 (2000)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)zbMATHGoogle Scholar
  71. 71.
    Orlik, P., Raymond, F.: Actions of the torus on a \(4\)-manifold—II. Topology 13, 89–112 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  72. 72.
    Reinhart, B.L.: Harmonic integrals on foliated manifolds. Am. J. Math. 81, 529–536 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Rukimbira, P.: Topology and closed characteristics of \(K\)-contact manifolds. Bull. Belg. Math. Soc. Simon Stevin 2, 349–356 (1995)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Shiga, H.: Equivariant de Rham cohomology of homogeneous spaces. J. Pure Appl. Algebra 106(2), 173–183 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Shiga, H., Takahashi, H.: Remarks on equivariant cohomology of homogeneous spaces. Technical Report 17, Technological University of Nagaoka (1995)Google Scholar
  76. 76.
    Skjelbred, T.: On the spectral sequence for the equivariant cohomology of a circle action. Preprint. Matematisk institutt, Universitetet i Oslo (1991)Google Scholar
  77. 77.
    Tymoczko, J.S.: An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson. Snowbird Lectures in Algebraic Geometry, Contemporary Mathematics, vol. 388, pp. 169–188 . American Mathematical Society, Providence (2005)Google Scholar
  78. 78.
    Varadarajan, V.S.: Lie groups, Lie algebras, and their representations. Reprint of the 1974 edition. Graduate Texts in Mathematics, vol. 102. Springer, New York (1984)Google Scholar
  79. 79.
    Weibel, C.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar

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© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikPhilipps Universität MarburgMarburgGermany

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