Goertsches, Oliver; Hagh Shenas Noshari, Sam; Mare, Augustin-Liviu

On the Equivariant Cohomology of Hyperpolar Actions on Symmetric Spaces

Doc. Math. 24, 1657-1676 (2019)
DOI: 10.25537/dm.2019v24.1657-1676
Communicated by Eckhard Meinrenken

Summary

We show that the equivariant cohomology of any hyperpolar action of a compact and connected Lie group on a symmetric space of compact type is a Cohen-Macaulay ring. This generalizes some results previously obtained by the authors.

Mathematics Subject Classification

53C35, 55N91, 57S15

Keywords/Phrases

symmetric spaces of compact type, hyperpolar actions, Hermann actions, equivariant cohomology, Cohen-Macaulay rings and modules

References

  • 1. J. F. Adams, Spin(8), triality, F4, and all that, Superspace and Supergravity, 435-445, ed. S. Hawking and M. Rocek, Cambridge Univ. Press, Cambridge, 1981. zbl 0465.53041 (Proceedings); MR0625301 (Proceedings).
  • 2. C. Allday, On the rank of a space, Trans. Amer. Math. Soc. 166 (1972), 173-185. DOI 10.2307/1996040; zbl 0239.57020; MR2620339.
  • 3. C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics, Vol. 32, Cambridge University Press, Cambridge, 1993. zbl 0799.55001; MR1236839.
  • 4. M. Atiyah, Elliptic Operators and Compact Groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin, 1974. DOI 10.1007/BFb0057821; zbl 0297.58009; MR0482866.
  • 5. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. DOI 10.1016/0040-9383(84)90021-1; zbl 0521.58025; MR0721448.
  • 6. T. J. Baird, Classifying spaces of twisted loop groups, Algebr. Geom. Topol. 16 (2016), no. 1, 211-229. DOI 10.2140/agt.2016.16.211; zbl 1337.22011; MR3470700; arxiv 1312.7450.
  • 7. A. Borel, Seminar in transformation groups, Annals of Math. Studies, Princeton University Press, Princeton, 1960. MR0116341.
  • 8. G. E. Bredon, The free part of a torus action and related numerical equalities, Duke Math. J. 41 (1974), 843-854. DOI 10.1215/S0012-7094-74-04184-2; zbl 0294.57024; MR0362376.
  • 9. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, Cambridge, 1993. zbl 0788.13005; MR1251956.
  • 10. J. D. Carlson, The Borel equivariant cohomology of real Grassmannians. Preprint. arxiv 1611.01175.
  • 11. J. D. Carlson, Equivariant formality of isotropic torus actions, I, J. Homotopy Relat. Struct. 14 (2019), no. 1, 199-234. DOI 10.1007/s40062-018-0207-5; zbl 07055729; MR3913974; arxiv 1410.5740.
  • 12. J. D. Carlson, On the Equivariant Cohomology of Homogeneous Spaces, www.math.toronto.edu/jcarlson/homogeneous_space_book.pdf (version of Jan. 10, 2019; accessed on March 7, 2019).
  • 13. J. D. Carlson and C.-K. Fok, Equivariant formality of isotropy actions, J. Lond. Math. Soc. 97 (2018), no. 3, 470-494. DOI 10.1112/jlms.12116; zbl 1401.55007; MR3816396; arxiv 1511.06228.
  • 14. P. E. Conner, On the action of the circle group, Michigan Math. J. 4 (1957), no. 3, 241-247. DOI 10.1307/mmj/1028997955; zbl 0081.39001; MR0096747.
  • 15. J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), 627-637. DOI 10.1007/BF02566231; zbl 0493.55003; MR0656216.
  • 16. M. Franz, Syzygies in equivariant cohomology for non-abelian Lie groups, pp. 325-360 in: Filippo Callegaro et al. (eds.), Configuration spaces (Cortona, 2014), Springer INdAM Ser. 14, Springer, Cham, 2016. DOI 10.1007/978-3-319-31580-5_14; zbl 1388.55007; MR3615739; arxiv 1409.0681.
  • 17. M. Franz and V. Puppe, Exact cohomology sequences with integral coefficients for torus actions, Transform. Groups 12 (2007), 65-76. DOI 10.1007/s00031-005-1127-0; zbl 05156780; MR2308029; arxiv math/0505607.
  • 18. M. Franz and V. Puppe, Exact sequences for equivariantly formal spaces, C. R. Math. Acad. Sci. Soc. R. Can. 33 (2011), 1-10. zbl 1223.55003; MR2790824; arxiv math/0307112.
  • 19. O. Goertsches, The equivariant cohomology of isotropy actions on symmetric spaces, Documenta Math. 17 (2012), 79-94. https://www.elibm.org/article/10000247; zbl 1247.53063; MR2889744; arxiv 1009.4079.
  • 20. O. Goertsches and S. Hagh Shenas Noshari, Equivariant formality of isotropy actions on homogeneous spaces defined by Lie group automorphisms, J. Pure Appl. Algebra 220 (2016), 2017-2028. DOI 10.1016/j.jpaa.2015.10.013; zbl 1335.53065; MR3437278; arxiv 1405.2655.
  • 21. O. Goertsches and A.-L. Mare, Non-abelian GKM theory, Math. Z. 277 (2014), 1-27. DOI 10.1007/s00209-013-1242-x; zbl 1305.57052; MR3205761; arxiv 1208.5568.
  • 22. O. Goertsches and A.-L. Mare, Equivariant cohomology of cohomogeneity one actions, Top. Appl. 167 (2014), 36-52. DOI 10.1016/j.topol.2014.03.006; zbl 1293.55006; MR3193423; arxiv 1110.6318.
  • 23. O. Goertsches and S. Rollenske, Torsion in equivariant cohomology and Cohen-Macaulay actions, Transform. Groups 16 (2011), 1063-1080. DOI 10.1007/s00031-011-9154-5; zbl 1254.55003; MR2852490; arxiv 1009.0120.
  • 24. O. Goertsches and D. Töben, Torus actions whose equivariant cohomology is Cohen-Macaulay, J. of Topology 3 (2010), 1-28. DOI 10.1112/jtopol/jtq025; zbl 1208.55005; MR2746339; arxiv 0912.0637.
  • 25. O. Goertsches and L. Zoller, Equivariant de Rham cohomology: theory and applications. Preprint. arxiv 1812.09511.
  • 26. C. Gorodski and F. Podesta, Homogeneity rank of real representations of compact Lie groups, J. Lie Theory 15 (2005), 63-77. zbl 1132.22014; MR2115228; arxiv math/0310251.
  • 27. V. W. Guillemin, V. L. Ginzburg, and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, Vol. 96, Amer. Math. Soc., Providence, RI, 2002. zbl 1197.53002; MR1929136.
  • 28. V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag, Berlin, 1999. zbl 0934.55007; MR1689252.
  • 29. S. Hagh Shenas Noshari, On the equivariant cohomology of isotropy actions, Doctoral Dissertation, University of Marburg, 2018.
  • 30. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, Rhode Island, 2001. zbl 0993.53002; MR1834454.
  • 31. E. Heintze, R. Palais, G. Thorbergsson, and C.-L. Terng, Hyperpolar actions and k-flat homogeneous spaces, J. Reine Angew. Math. 454 (1994), 163-179. zbl 0804.53074; MR1288683.
  • 32. E. Heintze, R. Palais, G. Thorbergsson, and C.-L. Terng, Hyperpolar actions on symmetric spaces, Geometry, Topology and Physics, International Press, Cambridge, MA, 1995, 214-245. zbl 0871.57035; MR1358619.
  • 33. R. Hermann, Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11 (1960), 544-546. DOI 10.2307/2034708; zbl 0098.36603; MR0124436.
  • 34. W.-Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 85, Springer-Verlag, New York-Heidelberg, 1975. zbl 0429.57011; MR0423384.
  • 35. F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton, 1984. zbl 0553.14020; MR0766741.
  • 36. S. Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958), 63-68. DOI 10.1017/S0027763000023497; zbl 0084.18202; MR0103508.
  • 37. A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), 571-612. DOI 10.1090/S0002-9947-01-02803-3; zbl 1042.53034; MR1862559.
  • 38. A. Kollross, Polar actions on symmetric spaces, J. Differential Geom. 77 (2007), 425-482. DOI 10.4310/jdg/1193074901; zbl 1139.53025; MR2362321; arxiv math/0506312.
  • 39. A. Kollross, Low cohomogeneity and polar actions on exceptional compact Lie groups, Transform. Groups 14 (2009), 387-415. DOI 10.1007/s00031-009-9052-2; zbl 1179.22014; MR2504928; arxiv 0804.1677.
  • 40. A. Kollross, Hyperpolar actions on reducible symmetric spaces, Transform. Groups 22 (2017), 207-228. DOI 10.1007/s00031-016-9384-7; zbl 1383.53041; MR3620771; arxiv 1412.2626.
  • 41. O. Loos, Symmetric Spaces II: Compact Spaces and Classification, Mathematics Lecture Notes Series, W. A. Benjamin, Inc., New York, 1969. zbl 0175.48601; MR0239006.
  • 42. M. Poulsen, Depth, detection and associated primes in the cohomology of finite groups (An introduction to Carlson's depth conjecture), Master Thesis, University of Copenhagen, 2007. web.math.ku.dk/~moller/students/mortenP.pdf.
  • 43. D. Quillen, The spectrum of an equivariant cohomology ring: I, Ann. Math. 94 (1971), 549-572. DOI 10.2307/1970770; zbl 0247.57013; MR0298694.
  • 44. J.-P. Serre, Alg\``ebre locale. Multiplicit\''es, 3 éd., Lecture Notes in Mathematics, Vol. 11, Springer-Verlag, Berlin, 1975. zbl 0296.13018.
  • 45. J. de Siebenthal, Sur les groupes de Lie compacts non connexes, Comm. Math. Helvetici 31 (1956), 41-89. DOI 10.1007/BF02564352; zbl 0075.01602; MR0094408.
  • 46. V. S. Varadarajan \(, Spin(7)\)-subgroups of \(SO(8)\) and \(Spin(8),\) Expo. Math. 19 (2001), 163-177. DOI 10.1016/S0723-0869(01)80027-X; zbl 0984.22010; MR1835966.

Affiliation

Goertsches, Oliver
Fachbereich Mathematik und Informatik, Philipps-Universit\``at Marburg, Hans-Meerwein-Str. 6, D-35043 Marburg, Germany
Hagh Shenas Noshari, Sam
Institut f\''ur Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Mare, Augustin-Liviu
Department of Mathematics and Statistics, University of Regina, 3737 Wascana Parkway, Regina, Canada S4S 0A2

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