From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$

Consequently, can we obtain a fibration of limit $$ S^2\to \mathbb{C}P^{\infty}\to\mathbb{H}P^\infty? $$


This is completely analogously to my answer of your other question by taking $G=Sp(1)=S^3, H=U(1)=S^1$.

One concludes that $$Sp(1)\rightarrow ESp(1)=S^{\infty}\rightarrow BSp(1)=\mathbb{H}P^{\infty}$$ is a model for the universal principal $Sp(1)$-bundle and the bundle you mention is just $$Sp(1)/U(1)=S^2\rightarrow BU(1)=\mathbb{C}P^{\infty}\rightarrow BSp(1)=\mathbb{H}P^{\infty}.$$

  • $\begingroup$ yes. thank you so much! $\endgroup$ – Shiquan Dec 3 '14 at 11:51

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