# Equivariant de Rham cohomology: theory and applications

- 50 Downloads

## Abstract

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.

## Keywords

Lie group actions Equivariant Cohomology Cartan model Equivariant formality Fixed points## Notes

### Acknowledgements

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.

## References

- 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.Allday, C., Franz, M., Puppe, V.: Equivariant cohomology, syzygies and orbit structure. Trans. Am. Math. Soc.
**366**, 6567–6589 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 3.Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar
- 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.Atiyah, M.F.: Elliptic Operators and Compact Groups. Lecture Notes in Mathematics, vol. 401. Springer, Berlin (1974)zbMATHCrossRefGoogle Scholar
- 6.Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc.
**14**(1), 1–15 (1982)MathSciNetzbMATHCrossRefGoogle Scholar - 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.Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969)zbMATHGoogle Scholar
- 9.Audin, M.: Torus Actions on Symplectic Manifolds, 2nd revised edn. Progress in Mathematics, vol. 93. Birkhäuser Verlag, Basel (2004)Google Scholar
- 10.Bazzoni, G., Goertsches, O.: \(K\)-cosymplectic manifolds. Ann. Glob. Anal. Geom.
**47**(3), 239–270 (2015)CrossRefMathSciNetzbMATHGoogle Scholar - 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.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.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.Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)zbMATHCrossRefGoogle Scholar
- 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.Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972)zbMATHGoogle Scholar
- 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.Carlson, J.D.: Equivariant formality of isotropic torus actions. J. Homotopy Relat. Struct.
**14**(1), 199–234 (2019)MathSciNetzbMATHCrossRefGoogle Scholar - 19.Carlson, J.D., Fok, C.-K.: Equivariant formality of isotropy actions. J. Lond. Math. Soc.
**97**(3), 470–494 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 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.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.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.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.Chang, T., Skjelbred, T.: The topological Schur lemma and related results. Ann. Math.
**2**(100), 307–321 (1974)MathSciNetzbMATHCrossRefGoogle Scholar - 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.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.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.Duflot, J.: Smooth toral actions. Topology
**22**(3), 253–265 (1983)MathSciNetzbMATHCrossRefGoogle Scholar - 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.Fok, C.-K.: Cohomology and \(K\)-theory of compact Lie groups (2018). Preprint. http://pi.math.cornell.edu/ckfok/. Accessed 5 Oct 2018
- 31.Franz, M.: Big polygon spaces. Int. Math. Res. Not. IMRN
**24**, 13379–13405 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 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.Franz, M., Puppe, V.: Exact cohomology sequences with integral coefficients for torus actions. Transform. Groups
**12**(1), 65–76 (2007)MathSciNetzbMATHCrossRefGoogle Scholar - 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.Ginzburg, V.: Equivariant cohomology and Kähler geometry. Funct. Anal. Appl.
**21**(4), 271–283 (1987).**(in Russian)**zbMATHCrossRefGoogle Scholar - 36.Goertsches, O.: The equivariant cohomology of isotropy actions on symmetric spaces. Doc. Math.
**17**, 79–94 (2012)MathSciNetzbMATHGoogle Scholar - 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.Goertsches, O., Hagh Shenas Noshari, S., Mare, A.-L.: On the equivariant cohomology of hyperpolar actions on symmetric spaces. Preprint. arXiv:1808.10630
- 39.Goertsches, O., Mare, A.-L.: Equivariant cohomology of cohomogeneity one actions. Topol. Appl.
**167**, 36–52 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 40.Goertsches, O., Mare, A.-L.: Non-abelian GKM theory. Math. Z.
**277**(1–2), 1–27 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 41.Goertsches, O., Nozawa, H., Töben, D.: Equivariant cohomology of K-contact manifolds. Math. Ann.
**354**(4), 1555–1582 (2012)MathSciNetzbMATHCrossRefGoogle Scholar - 42.Goertsches, O., Rollenske, S.: Torsion in equivariant cohomology and Cohen-Macaulay \(G\)-actions. Transform. Groups
**16**(4), 1063–1080 (2011)CrossRefMathSciNetzbMATHGoogle Scholar - 43.Goertsches, O., Töben, D.: Torus actions whose cohomology is Cohen-Macaulay. J. Topol.
**3**(4), 819–846 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - 44.Goertsches, O., Töben, D.: Equivariant basic cohomology of Riemannian foliations. J. Reine Angew. Math.
**745**, 1–40 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 45.Goertsches, O., Wiemeler, M.: Positively curved GKM-manifolds. Int. Math. Res. Not. IMRN
**22**, 12015–12041 (2015)MathSciNetzbMATHGoogle Scholar - 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.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.Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra
**91**(1–3), 137–142 (1994)MathSciNetzbMATHCrossRefGoogle Scholar - 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.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.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.Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
- 53.Guillemin, V., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Mathematics Past and Present. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
- 54.Guillemin, V., Zara, C.: Equivariant de Rham theory and graphs. Asian J. Math
**3**(1), 49–76 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 55.Hagh Shenas Noshari, S.: On the equivariant cohomology of isotropy actions. Dissertation, Philipps University of Marburg (2018)Google Scholar
- 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.He, C.: Localization of certain odd-dimensional manifolds with torus actions. Preprint. arXiv:1608.04392
- 58.Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1975)CrossRefGoogle Scholar
- 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.Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press, Princeton (1984)zbMATHGoogle Scholar
- 61.Knapp, A.W.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser Boston, Inc., Boston (2002)zbMATHGoogle Scholar
- 62.Kobayashi, S.: Fixed points of isometries. Nagoya Math. J.
**13**, 63–68 (1958)MathSciNetzbMATHCrossRefGoogle Scholar - 63.Kumar, S.: Kac-Moody Groups, Their Flag Varieties, and Representation Theory. Birkhäuser, Basel (2001)Google Scholar
- 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). http://uregina.ca/~mareal/flag-coh.pdf. Accessed 5 Oct 2018
- 65.McCleary, J.: A User’s Guide to Spectral Sequences. Cambridge University Press, Cambridge (2000)zbMATHCrossRefGoogle Scholar
- 66.Meinrenken, E.: Equivariant Cohomology and the Cartan Model, Encyclopedia of Mathematical Physics. Elsevier, Amsterdam (2006)Google Scholar
- 67.Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
- 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.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.Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)zbMATHGoogle Scholar
- 71.Orlik, P., Raymond, F.: Actions of the torus on a \(4\)-manifold—II. Topology
**13**, 89–112 (1974)CrossRefMathSciNetzbMATHGoogle Scholar - 72.Reinhart, B.L.: Harmonic integrals on foliated manifolds. Am. J. Math.
**81**, 529–536 (1959)MathSciNetzbMATHCrossRefGoogle Scholar - 73.Rukimbira, P.: Topology and closed characteristics of \(K\)-contact manifolds. Bull. Belg. Math. Soc. Simon Stevin
**2**, 349–356 (1995)MathSciNetzbMATHGoogle Scholar - 74.Shiga, H.: Equivariant de Rham cohomology of homogeneous spaces. J. Pure Appl. Algebra
**106**(2), 173–183 (1996)MathSciNetzbMATHCrossRefGoogle Scholar - 75.Shiga, H., Takahashi, H.: Remarks on equivariant cohomology of homogeneous spaces. Technical Report 17, Technological University of Nagaoka (1995)Google Scholar
- 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.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.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.Weibel, C.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar