I need help on constructing the fiber bundles $S^3 \rightarrow S^{7} \rightarrow \mathbb{H}P^1$ I heard that you just use the same idea as the hopf map to $\mathbb{C}P^n$.
So I guess you say $S^{7}$ are the vectors in $\mathbb{H}^{2}$. But, then what does would that mean. Also, do I then take a quotient $q: S^{7} \rightarrow \mathbb{H}P^1$.
Also, I'm a bit confused on showoing $q^-1(U_a) \cong U_a \times S^3$. I assume I have to prove this and then give the manifolds for it to prove that it's a fiber bundle.
Anyone know a decent paper that explain this Hopf map?