A GKM description of the equivariant cohomology ring of a homogeneous space
V. Guillemin1
, T. Holm2
and C. Zara2
1Department of Mathematics, MIT, Cambridge, MA 02139
2Department of Mathematics, Penn State Altoona, PA, 16601
2Department of Mathematics, Penn State Altoona, PA, 16601
DOI: 10.1007/s10801-006-6027-4
Abstract
Let T be a torus of dimension n > 1 and M a compact T-manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/ T is zero dimensional and the set of one dimensional orbits in M/ T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph.
In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.
Pages: 21–41
Keywords: keywords GKM graph; homogeneous spaces; equivariant cohomology
Full Text: PDF
References
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2. A. Borel and J. De Siebenthal, “Les sous-groupes fermés de rang maximum des groupes de Lie clos (French),” Comment. Math. Helv. 23 (1949), 200-221.
3. W. Fulton and J. Harris, Representation theory, Springer, New York, 1991.
4. W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, vol. II, Academic Press, 1973.
5. W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, vol. III, Academic Press,
1976. Springer
6. M. Goresky, R. Kottwitz and R. MacPherson, “Equivariant cohomology, Koszul duality, and the localiza- tion theorem,” Invent. Math. 131(1) (1998), 25-83.
7. R. Goldin, The cohomology of weight varieties, Ph.D. thesis, MIT 1999.
8. V. Guillemin and S. Sternberg, Supersymetry and equivariant de Rham cohomology, Springer Verlag, Berlin, 1999.
9. V. Guillemin and C. Zara, “One-skeleta, Betti numbers and equivariant cohomology,” Duke Math. J. 107(2) (2001), 283-349.
10. J. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, 1990.
11. T. Holm, “Homogeneous spaces, equivariant cohomology, and graphs,” Ph.D. thesis, MIT 2002.
12. N. Steenrod, The topology of fibre bundles, Princeton University Press, 1951.
13. D. Vogan, personal communication.
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