## Nonexistence of almost complex structures on Grassmann manifolds

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- 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