Grassmannians and the equivariant cohomology of isotropy actions
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- by Jeffrey D. Carlson PDF
- Proc. Amer. Math. Soc. 149 (2021), 4469-4485
Abstract:
We prove a general structure theorem for the rational Borel equivariant cohomology ring of an equivariantly formal isotropy action, rederiving He’s recent computation of the equivariant cohomology of real Grassmannians as an illustration.References
- Paul F. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15–38. MR 219085, DOI 10.1016/0040-9383(86)90012-1
- Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728
- Edgar H. Brown Jr., The cohomology of $B\textrm {SO}_{n}$ and $B\textrm {O}_{n}$ with integer coefficients, Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288. MR 652459, DOI 10.1090/S0002-9939-1982-0652459-1
- Jeffrey D. Carlson, The Borel equivariant cohomology of real Grassmannians, 2016. arXiv:1611.01175.
- Jeffrey D. Carlson, Equivariant formality of isotropic torus actions, J. Homotopy Relat. Struct. 14 (2019), no. 1, 199–234. MR 3913974, DOI 10.1007/s40062-018-0207-5
- Jeffrey D. Carlson and Chi-Kwong Fok, Equivariant formality of isotropy actions, J. Lond. Math. Soc. (2) 97 (2018), no. 3, 470–494. MR 3816396, DOI 10.1112/jlms.12116
- Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 57–71 (French). MR 0042427
- Luis Casian and Yuji Kodama, On the cohomology of real Grassmann manifolds, arXiv:1309.5520, 2013.
- J.-H. Eschenburg, Cohomology of biquotients, Manuscripta Math. 75 (1992), no. 2, 151–166. MR 1160094, DOI 10.1007/BF02567078
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Yves Félix, John Oprea, and Daniel Tanré, Algebraic models in geometry, Oxford Graduate Texts in Mathematics, vol. 17, Oxford University Press, Oxford, 2008. MR 2403898
- Matthias Franz, The cohomology rings of homogeneous spaces, arXiv:1907.04777, 2019.
- Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
- Oliver Goertsches, The equivariant cohomology of isotropy actions on symmetric spaces, Doc. Math. 17 (2012), 79–94. MR 2889744
- Oliver Goertsches and Sam Haghshenas Noshari, Equivariant formality of isotropy actions on homogeneous spaces defined by Lie group automorphisms, J. Pure Appl. Algebra 220 (2016), no. 5, 2017–2028. MR 3437278, DOI 10.1016/j.jpaa.2015.10.013
- V. K. A. M. Gugenheim and J. Peter May, On the theory and applications of differential torsion products, Memoirs of the American Mathematical Society, No. 142, American Mathematical Society, Providence, R.I., 1974. MR 0394720
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Allen Hatcher, Spectral sequences in algebraic topology, 2004 manuscript. http://math.cornell.edu/~hatcher/SSAT/SSATpage.html.
- Allen Hatcher, Vector bundles and K-theory, 2017 manuscript. http://math.cornell.edu/~hatcher/VBKT/VBpage.html.
- Chen He, GKM theory, characteristic classes and the equivariant cohomology ring of the real Grassmannian, arXiv:1609.06243, 2016.
- Heinz Hopf, Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann. of Math. (2) 42 (1941), 22–52 (German). MR 4784, DOI 10.2307/1968985
- Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384, DOI 10.1007/978-3-642-66052-8
- Vitali Kapovitch, A note on rational homotopy of biquotients, 2004. http://www.math.toronto.edu/vtk/biquotient.pdf.
- Shrawan Kumar and Michèle Vergne, Equivariant cohomology with generalized coefficients, Astérisque 215 (1993), 109–204. Sur la cohomologie équivariante des variétés différentiables. MR 1247061
- Jean Leray, Détermination, dans les cas non exceptionnels, de l’anneau de cohomologie de l’espace homogène quotient d’un groupe de Lie compact par un sous-groupe de même rang, C. R. Acad. Sci. Paris 228 (1949), 1902–1904 (French). MR 30961
- Clair E. Miller, The topology of rotation groups, Ann. of Math. (2) 57 (1953), 90–114. MR 52772, DOI 10.2307/1969727
- Hans J. Munkholm, The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure Appl. Algebra 5 (1974), 1–50. MR 350735, DOI 10.1016/0022-4049(74)90002-4
- Arkadi L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. MR 1266842
- Rustam Sadykov, Elementary calculation of the cohomology rings of real Grassmann manifolds, Pacific J. Math. 289 (2017), no. 2, 443–447. MR 3667179, DOI 10.2140/pjm.2017.289.443
- Hiroo Shiga, Equivariant de Rham cohomology of homogeneous spaces, J. Pure Appl. Algebra 106 (1996), no. 2, 173–183. MR 1372850, DOI 10.1016/0022-4049(95)00018-6
- W. Singhof, On the topology of double coset manifolds, Math. Ann. 297 (1993), no. 1, 133–146. MR 1238411, DOI 10.1007/BF01459492
- Masaru Takeuchi, On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1962), 313–328 (1962). MR 145009
- Emery Thomas, On the cohomology of the real Grassmann complexes and the characteristic classes of $n$-plane bundles, Trans. Amer. Math. Soc. 96 (1960), 67–89. MR 121800, DOI 10.1090/S0002-9947-1960-0121800-0
- Loring W. Tu, Computing characteristic numbers using fixed points, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 185–206. MR 2648896, DOI 10.1090/crmp/050/19
- Joel Wolf, The cohomology of homogeneous spaces, Amer. J. Math. 99 (1977), no. 2, 312–340. MR 438370, DOI 10.2307/2373822
Additional Information
- Jeffrey D. Carlson
- Affiliation: Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom
- MR Author ID: 985506
- Email: j.carlson@imperial.ac.uk
- Received by editor(s): April 6, 2019
- Published electronically: July 16, 2021
- Communicated by: Mark Behrens
- © Copyright 2021 Copyright by the author
- Journal: Proc. Amer. Math. Soc. 149 (2021), 4469-4485
- MSC (2020): Primary 55N25, 57T15
- DOI: https://doi.org/10.1090/proc/14906
- MathSciNet review: 4305997