Nonexistence of almost complex structures on Grassmann manifolds
HTML articles powered by AMS MathViewer
- by Parameswaran Sankaran PDF
- Proc. Amer. Math. Soc. 113 (1991), 297-302 Request permission
Abstract:
In this paper we prove that, for $3 \leq k \leq n - 3$, none of the oriented Grassmann manifolds, ${\widetilde {G}_{n,k}}$—except for ${\widetilde {G}_{6,3}}$, and a few as yet undecided cases—admits a weakly almost complex structure. The result for $k = 1,2,n - 1,n - 2$ are well known and classical. The proofs make use of basic concepts in $K$-theory, the property that ${\widetilde {G}_{n,k}}$ is $(n - k)$-universal, known facts about $K(\mathbb {H}{P^4})$, and characteristic classes.References
- J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
- M. F. Atiyah and J. A. Todd, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 56 (1960), 342–353. MR 132552, DOI 10.1017/s0305004100034642
- Vojtěch Bartík and Július Korbaš, Stiefel-Whitney characteristic classes and parallelizability of Grassmann manifolds, Proceedings of the 12th winter school on abstract analysis (Srní, 1984), 1984, pp. 19–29. MR 782702
- A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
- A. Borel and J.-P. Serre, Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409–448 (French). MR 58213, DOI 10.2307/2372495
- W. C. Hsiang and R. H. Szczarba, On the tangent bundle of a Grassman manifold, Amer. J. Math. 86 (1964), 698–704. MR 172304, DOI 10.2307/2373153
- Dale Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Mathematics, No. 20, Springer-Verlag, New York-Heidelberg, 1975. MR 0370578
- Július Korbaš, On the nonexistence of almost complex structures on some oriented flag manifolds, Differential geometry and its applications, communications (Brno, 1986) Univ. J. E. Purkyně, Brno, 1987, pp. 175–179. MR 923378
- Kee Yuen Lam, A formula for the tangent bundle of flag manifolds and related manifolds, Trans. Amer. Math. Soc. 213 (1975), 305–314. MR 431194, DOI 10.1090/S0002-9947-1975-0431194-X
- Martin Markl, Note on the existence of almost complex structures on compact manifolds, Ann. Global Anal. Geom. 4 (1986), no. 2, 263–269. MR 910554, DOI 10.1007/BF00129911
- W. S. Massey, Non-existence of almost-complex structures on quaternionic projective spaces, Pacific J. Math. 12 (1962), 1379–1384. MR 151988
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
- P. Sankaran and P. Zvengrowski, Stable parallelizability of partially oriented flag manifolds, Pacific J. Math. 128 (1987), no. 2, 349–359. MR 888523
- Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
- Július Korbaš, Note on Stiefel-Whitney classes of flag manifolds, Proceedings of the Winter School on Geometry and Physics (Srní, 1987), 1987, pp. 109–111. MR 946716
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 297-302
- MSC: Primary 57R15; Secondary 57R20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1043420-1
- MathSciNet review: 1043420