Cohomology of GKM fiber bundles
Victor Guillemin
, Silvia Sabatini
and Catalin Zara
DOI: 10.1007/s10801-011-0292-6
Abstract
The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.
Pages: 19–59
Keywords: keywords equivariant fiber bundle; equivariant cohomology; GKM space; flag manifold
Full Text: PDF
References
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2. Billey, S.: Kostant polynomials and the cohomology ring for G/B. Duke Math. J. 96(1), 205-224 (1999)
3. Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math., 243(1-3), 21-66 (2002)
4. Chang, T., Skjelbred, T.: The topological Schur lemma and related results. Ann. Math. (2) 100, 307- 321 (1974)
5. Fulton, W., Harris, J.: Representation Theory. Springer, New York (1991)
6. Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25-83 (1998)
7. Guillemin, V., Holm, T.: GKM theory for torus actions with nonisolated fixed points. Int. Math. Res. Not. 40, 2105-2124 (2004)
8. Guillemin, V., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Mathematics Past and Present. Springer, Berlin (1999)
9. Guillemin, V., Zara, C.: Equivariant DeRham cohomology and graphs. Asian J. Math. 3(1), 49-76 (1999)
10. Guillemin, V., Zara, C.: 1-skeleta, Betti numbers, and equivariant cohomology. Duke Math. J. 107(2), 283-349 (2001)
11. Guillemin, V., Lerman, E., Sternberg, S.: Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press, Cambridge (1996)
12. Guillemin, V., Holm, T., Zara, C.: A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebr. Comb. 23(1), 21-41 (2006)
13. Guillemin, V., Sabatini, S., Zara, C.: Balanced fiber bundles and GKM theory (in preparation)
14. Hiller, H.: Schubert calculus of a Coxeter group. Enseign. Math. 27, 57-84 (1981)
15. Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol.
29. Cambridge University Press, Cambridge (1990)
16. Knutson, A.: A Schubert calculus recurrence from the noncomplex W-action on G/B.
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