Nonexistence of almost complex structures on Grassmann manifolds

Sankaran, Parameswaran

doi:10.1090/S0002-9939-1991-1043420-1

Nonexistence of almost complex structures on Grassmann manifolds
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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.

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