A note on the quaternionic quasi-projective space.(English)Zbl 0563.55004

According to I. M. James [The topology of Stiefel manifolds, Lond. Math. Soc. Lect. Note Ser. 24 (1976; Zbl 0337.55017)], the quaternionic quasi-projective space $${\mathbb{H}}{\mathbb{Q}}_ n$$ is defined in two ways. In this paper the authors show that the two definitions are equivalent and that the map $$t_ n: {\mathbb{H}}{\mathbb{Q}}_ n\to E({\mathbb{C}}{\mathbb{P}}_+^{2n- 1})$$ of H. Toda and K. Kozima [J. Math. Kyoto Univ. 22, 131- 153 (1982; Zbl 0502.55004)] can be identified with the connecting map in the cofibre sequence starting with $$g_{n+}: {\mathbb{C}}{\mathbb{P}}_+^{2n- 1}\to {\mathbb{H}}{\mathbb{P}}_+^{n-1}$$. Here, $$X_+=X\cup \{one$$ point$$\}$$ and $$g_{n+}$$ stands for the map induced from the standard fibration $$g_ n: {\mathbb{C}}{\mathbb{P}}^{2n-1}\to {\mathbb{H}}{\mathbb{P}}^{n-1}$$. The authors also construct a natural mapping from $$E({\mathbb{H}}{\mathbb{P}}_+^{n-1})$$ to $$X_ n=U(2n)/Sp(n)$$ by use of the first result. As an application, the authors determine the order of the attaching class $$E\gamma_{n- 1}({\mathbb{H}})\in \pi_{4n}(E{\mathbb{H}}{\mathbb{P}}^{n-1})$$ for even n. [Dr. Shichirô Oka died on 30th October, 1984.]

MSC:

 55P05 Homotopy extension properties, cofibrations in algebraic topology 22E15 General properties and structure of real Lie groups 55R05 Fiber spaces in algebraic topology

Citations:

Zbl 0337.55017; Zbl 0502.55004
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