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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonexistence of almost complex structures on Grassmann manifolds


Author: Parameswaran Sankaran
Journal: Proc. Amer. Math. Soc. 113 (1991), 297-302
MSC: Primary 57R15; Secondary 57R20
MathSciNet review: 1043420
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1043420-1
PII: S 0002-9939(1991)1043420-1
Article copyright: © Copyright 1991 American Mathematical Society