Nonexistence of almost complex structures on Grassmann manifolds

Abstract

In this paper we prove that, for 3≤k≤n-3, none of the oriented Grassmann manifolds, G_n,k--except for 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 G_n,k is (n-k)-universal, known facts about K(HP4), and characteristic classes.