On Quaternionic James Numbers and Almost-Quaternion Substructures on the Sphere

Almost-Quaternion Substructures on the Canonical Cn−1 -Bundle over S2n−1

“…A similar problem, the existence problem of almost complex substructures, was solved by I. Dibag in [2], and the existence problem of almost-quaternion /c-substructures on spheres was solved in our paper [5]. The techniques and results of these papers together with some techniques used in [7] yielded the solution of the present problem.…”

“…?2n-1 has a cross section. It is clear that an almost-quaternion /c-substructure on ^n_ 1 gives rise to an almost-quaternion /c-substructure on S 2 "" 1 , in the sense of [5]. Another interpretation of an almost-auaternion /c-substructure on ^n_ i can be given as follows.…”

“…We shall denote by x nk the element of 7r 4n _ 2 (A r n _ 1 >fc -i) which is the obstruction to cross sectioning the symplectic Stiefel fibering X n k -> S 4 "" 1 , and let us denote by X{n,k} the order of x nfc in Proof. The proof of this lemma is almost the same as that of Lemma 3.2 of [5].…”

“…To prove this theorem we shall use techniques from [2,5]. Let X nk = Sp(n)/Sp(w-fe), and let 4n-1 denote the symplectic Stiefel fibering.…”

A Remark on Almost-Quaternlon Substructures on the Sphere

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In [4] T. Onder has solved the existence problem of almostquaternion ^-substructures on the ^-sphere S" for all n and k except for n = l (mod 4)^5 and k=(n -l)/4. The purpose of this note is to solve it for this exceptional case.

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