Abstract
We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.
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Notes
Note that there is a slight error in [7, Proposition 4.2]: on the right hand side of the equation one has to consider the subgroup of \(N_G(\mathfrak {t}_p)\) consisting of those elements which leave invariant \(M^{\mathfrak {t}_p,p}\) instead of the whole normalizer \(N_G(\mathfrak {t}_p)\).
References
Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Cambridge (1993)
Bredon, G.E.: The free part of a torus action and related numerical equalities. Duke Math. J. 41, 843–854 (1974)
Chang, T., Skjelbred, T.: The topological Schur lemma and related results. Ann. Math. 100, 307–321 (1974)
Franz, M., Puppe, V.: Exact sequences for equivariantly formal spaces. C. R. Math. Acad. Sci. Soc. R. Can. 33(1), 1–10 (2011)
Goertsches, O., Mare, A.-L.: Equivariant cohomology of cohomogeneity-one actions. preprint, arXiv:1110.6310 (2011)
Goertsches, O., Nozawa, H., Töben, D.: Equivariant cohomology of \(K\)-contact manifolds. Math. Ann. 354(4), 1555–1582 (2012)
Goertsches, O., Rollenske, S.: Torsion in equivariant cohomology and Cohen-Macaulay actions. Transform. Groups 16(4), 1063–1080 (2011)
Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent Math. 131(1), 25–83 (1998)
Guillemin, V.W., Ginzburg, V.L., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions, Mathematical Surveys and Monographs, vol. 96. AMS, Providence, RI (2002)
Guillemin, V. W., Holm, T.: GKM theory for torus actions with nonisolated fixed points. Int. Math. Res. Not. 2004 (40), 2105–2124 (2004)
Guillemin, V.W., Holm, T., Zara, C.: A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebraic Comb. 23(1), 21–41 (2006)
Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999)
Guillemin, V.W., Zara, C.: Equivariant de Rham theory and graphs. Asian J. Math. 3, 49–76 (1999)
Guillemin, V.W., Zara, C.: One-skeleta, Betti numbers, and equivariant cohomology. Duke Math. J. 107, 283–349 (2001)
Harada, M., Henriques, A., Holm, T.: Computation of generalized equivariant cohomologies of Kac-Moody flag varieties. Adv. Math. 197, 198–221 (2005)
Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pacific J. Math. 246(1), 129–185 (2010)
Hsiang, W.-Y.: Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 85. Springer, New York (1975)
Kuroki, S.: Characterization of homogeneous torus manifolds. Osaka J. Math. 47, 285–299 (2010)
Kuroki, S.: Classification of torus manifolds with codimension one extended actions. Transform. Groups 16, 481–536 (2011)
Kuroki, S.: Introduction to GKM theory, Proceedings of ASARC Workshop, Daecheon & Muju 2009, Trends in Mathematics, 11(2), 113–129 (2009)
Mare, A.-L.: Equivariant cohomology of quaternionic flag manifolds. J. Algebra 319(7), 2830–2844 (2008)
Mare, A.-L., Willems, M.: Topology of the octonionic flag manifold preprint, arXiv:0809.4318, to appear in Münster J. Math. (2008)
Masuda, M.: Symmetry of a symplectic toric manifold. J. Symplectic Geom. 8, 359–380 (2010)
Masuda, M., Panov, T.: On the cohomology of torus manifolds. Osaka J. Math. 43, 711–746 (2006)
Mostert, P.S.: On a compact Lie group acting on a manifold. Ann. Math. 65(3), 447–455 (1957)
Wiemeler, M.: Torus manifolds with non-abelian symmetries. Trans. Amer. Math. Soc. 364, 1427–1487 (2012)
W. Ziller.: On the geometry of cohomogeneity one manifolds with positive curvature. In: Riemannian Topology and Geometric Structures on Manifolds. Progr. Math., vol. 271, pp. 233–262. Birkhäuser Boston, Boston, MA (2009)
Acknowledgments
We wish to thank the anonymous referee for suggesting several improvements.