Group Actions on Flag Manifolds and Cobordism

Sankaran, P. ; Varadarajan, K.

Canadian journal of mathematics, Tome 45 (1993) no. 3, pp. 650-661

Résumé

In this paper we show that any (respectively α є Ωm ) can be represented by a closed smooth (respectively closed, oriented smooth) manifold Mm admitting a smooth (Z/2)m (respectively S 1)-action with a finite stationary set. We also completely determine the Grassman manifolds which are oriented boundaries as well as those which represent non-torsion elements in Ω*.

Détail

DOI : 10.4153/CJM-1993-036-8

Mots-clés : 57R75, 57R85, (Oriented) cobordism, stationary set, Pontrjagin numbers

Bibliographie

[1] 1. Adams, J. F., Lectures on Lie Groups, University of Chicago Press, Midway Reprint, (1982). Google Scholar

[2] 2. Borel, A., La cohomologie mod2 de certains espaces homogènes, Comm. Math. Helv. 27(1953), 165— 197. Google Scholar

[3] 3. Borel, A., Sur la cohomologie des espaces fibres principaux et des expaces homogènes de groupe de Lie compacts, Ann. Math. 57(1953), 115–207. Google Scholar

[4] 4. Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces, Amer. J. Math. 80(1958), 458–535. Google Scholar

[5] 5. Chern, S. S., Hirzebruch, F., and J.-R Serre, The index of a fibered manifold, Proc. Amer. Math. Soc. 8(1957), 587–596. Google Scholar

[6] 6. Conner, P. E., Differentiable periodic maps, 2nd Ed., L.N.M. (738), Springer-Verlag, 1979. Google Scholar

[7] 7. Conner, P. E. and Floyd, E. E., Differentiable periodic maps, Ergebnisse Sr-33, Springer-Verlag, 1964. Google Scholar

[8] 8. Floyd, E. E., Stiefel-Whitney numbers of quaternionic and related manifolds, Trans. Amer. Math. Soc. 155(1971), 77–94. Google Scholar

[9] 9. Hsiang, W.-C. and Szczarba, R. H., On the tangent bundle for the Grassmann manifold, Amer. J. Math. 86(1964), 698–704. Google Scholar

[10] 10. Husemoller, D., Fibre bundles, G.T.M.(20), Springer-Verlag, 1985. Google Scholar

[11] 11. Kobayashi, S. and Nomizu, K., Foundations of differential geometry Vol. II, Interscience Publishers, 1969. Google Scholar

[12] 12. Korbas, J., Vector fields on real flag manifolds, Ann. Glob. Anal. Geom. 3(1985), 173–184. Google Scholar

[13] 13. Lam, K. Y., A formula for the tangent bundle of flag manifolds and related manifolds, Trans. Amer. Math. Soc. 213(1975), 305–314. Google Scholar

[14] 14. Milnor, J. W., On the cobordism ring Ω and a complex analogue, Amer. J. Math. 82(1960), 505–521. Google Scholar

[15] 15. Milnor, J. W. and Stasheff, J. D., Characteristic classes, Ann. Math. Stud. 76, Princeton, (1974). Google Scholar

[16] 16. Mong, S., The index of complex and quaternionic Grassmannians via Lefschetz formula, Adv. Math. 15(1975), 169–174. Google Scholar

[17] 17. Sankaran, P., Cobordism of flag manifolds, Institute of Mathematical Sciences, Madras, India, (1987), preprint. Google Scholar

[18] 18. Sankaran, P., Determination of Grassmann manifolds which are boundaries, Canad. Math. Bull. 34(1991), 119–122. Google Scholar

[19] 19. Sankaran, P., Which Grassmannians bound?, Arch. Math. 50(1988), 474–476 Google Scholar

[20] 20. Shanahan, P., On the signature ofGrassmannians, Pac. J. Math. 84(1979), 483–490. Google Scholar

[21] 21. Stewart, B. M., Theory of numbers, 2nd Ed. Macmillan & Co., New York, 1964. Google Scholar

[22] 22. Stong, R. E., Equivariant bordism and (Z)K actions, Duke. Math. J. 37(1970), 779–785. Google Scholar

[23] 23. Stong, R. E., Stationary point free group actions, Proc. Am. Math. Soc. 18(1967), 1089–1092. Google Scholar

[24] 24. Stong, R. E., Notes on cobordism theory, Mathematical Notes, Princeton Univ. Press, (1968). Google Scholar

[25] 25. Stong, R. E., Math. Reviews, 89d:57050. Google Scholar