Show that the fiber bundle \(S^3 \to S^{4n+3} \to \mathbb{HP}^n\) gives rise to a quotient fiber bundle \(S^2 \to \mathbb{CP}^{2n+1} \to \mathbb{HP}^n\) by factoring out the action of \(S^1\) on \(S^{4n+3}\) by complex scalar multiplication.
Solution
View \[ S^{4n+3}\subset\mathbb H^{n+1}. \] The quaternionic Hopf bundle is the quotient by right multiplication of unit quaternions: \[ S^3\longrightarrow S^{4n+3}\longrightarrow\mathbb{HP}^n. \] Inside \(S^3\) is the subgroup \[ S^1=\{e^{i\theta}\}\subset\mathbb C\subset\mathbb H. \] Its action on \(S^{4n+3}\) is precisely complex scalar multiplication after identifying \[ \mathbb H^{n+1}\cong\mathbb C^{2n+2}. \] Therefore \[ S^{4n+3}/S^1\cong\mathbb{CP}^{2n+1}. \]
Although \(S^1\subset S^3\) is not a normal subgroup, the quotient is taken fiberwise by the right \(S^1\)-action: over a quaternionic line \(L\), the Hopf fiber is \(S^3\), and dividing that fiber by the right \(S^1\)-action gives \[ S^3/S^1\cong S^2. \] The original local trivializations of the quaternionic Hopf bundle are \(S^3\)-equivariant, hence descend after quotienting by \(S^1\). Thus we obtain a locally trivial bundle \[ \boxed{S^2\longrightarrow\mathbb{CP}^{2n+1}\longrightarrow\mathbb{HP}^n.} \]