The total space of the standard quaternionic Hopf fibration of an $S^3$ fiber bundle over $S^4.$

Question

I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract:

I do not understand why the total space of this fiber bundle is given by:

Could anyone explain this also to me please?

Answer 1

By reading the lines that follow where the pre images of $U_1$ and $U_2$ are computed, I believe that there's a slight inaccuracy in the definition of the total space.

In section 2, McEnroe defines the Quaternionic Hopf Fibration as the map $\pi \colon \mathbb{S}^7 \subset \mathbb{H}^2 \to \mathbb{S}^4 = \mathbb{HP}^1$ given by $(x,y)\mapsto [x;y]$ and the total space is $\mathbb{S}^7$.

Now in section 3, it seems that instead of getting $\mathbb{S}^7$ itself as the total space, they consider the graph of $\pi$, that is, $$ \{ ((x,y),\pi(x,y)) \ : \ (x,y) \neq (0,0) \} = \{ ((x,y),[z;w]) \ : \ x = \lambda z, y = \lambda w \ \text{for some} \ \lambda \in \mathbb{H} \} \subset \mathbb{H}^2 \times \mathbb{HP}^1 $$ which is also the total space, since a given space is homeomorphic to the graph of any map coming from it (here $\pi$ is actually considered as having $\mathbb{H}^2\setminus \{(0,0)\}$ as its domain, but it's unimportant for now). Notice that in this representation, $\pi$ is simply the projection on the second coordinate $\mathbb{HP}^1$.

This characterization of the total space makes sense when considering the lines that follow in the article and if we were to try to naively simplify the expression defining this set, we could get something like $$ \frac{x}{z} = \lambda = \frac{y}{w} \iff xw = zy$$ which (at least in the way I've done it) is incorrect since multiplication isn't commutative and we were assuming both $z,w$ nonzero.