I'm finally at the end of Milnor's "On manifolds homeomorphic to the 7-sphere", and I stumbled upon something I cant figure out…
For those with the reference I'm talking about "lemma 5", it goes something like this, you have two $\mathbb{S}^3$ bundles over $\mathbb{S}^4$, we want to obtain the total space of this bundle, so you glue them via the transition function, one can think of this as having a pair of copies of $(\mathbb{R}^4 \setminus \{0\}) \times \mathbb{S}^3$ and gluing them by identifiying $(u,v) \mapsto (u',v')=(u / \|u\|^2, u^hvu^j/\|u\|)$ where $u$ and $v$ are quaternions, so far so good, now Milnor states that if $h+j =1$ then this manifold is a $7$-sphere, his reason is that the function $f(x) = \mathfrak{R}(v)/(1+\|u\|^2)^{1/2}$ is a morse function, this with the "first" coordinate chart, for the second he defines $u'' = u'(v')^{-1}$ and substitutes $(u',v')$ for $(u'',v')$ stating that the function $f$ is now given by $\mathfrak{R}(u'')/(1+\|u''\|^2)^{1/2}$.
He then says "It is easily verified that f has only two critical points (namely $(u,v) = \pm (0,1)$) and that these are nondegenerate".
That's where I get lost; I don't understand his change of coordinates $(u',v') \mapsto (u'',v')$, nor why he states the function is now the one stated… I tried developing the algebra but I can't get it to work out, I thought maybe he was using the involution $v \mapsto v^{-1}$ somehow but it doesn't add up either…
Best Answer
Milnor didn't explain the formula as much as maybe he should have, but the point is that the real part of a unit-length quaternion is invariant under both conjugation and inversion. Let $$r = ||u|| \qquad \hat{u} = u/r,$$ so that $$v' = \hat{u}^h v \hat{u}^j \qquad u' = \hat{u}r \qquad ||u'|| = ||u''|| = 1/r.$$ Thus $$v' \hat{u}^{-1} = \hat{u}^h v \hat{u}^{-h}$$ is conjugate to $v$. Thus $$\mathfrak{R}(v) = \mathfrak{R}(v'\hat{u}^{-1}) = \mathfrak{R}(\hat{u} (v')^{-1}).$$ The first equality is conjugation, the second one is inversion. So then you get $$\frac{\mathfrak{R}(v)}{\sqrt{1 + ||u||^2}} = \frac{\mathfrak{R}(v)}{\sqrt{1+r^2}} = \frac{\mathfrak{R}(\hat{u} (v')^{-1})}{\sqrt{1+r^2}} = \frac{\mathfrak{R}((\hat{u}/r) (v')^{-1})}{\sqrt{1+r^{-2}}} = \frac{\mathfrak{R}(u'')}{\sqrt{1+||u''||^2}}.$$ Note that, although $(u'',v')$ certainly is a valid parameterization of the second chart, it's enough to think of $u''$ as a convenient function rather than part of a coordinate frame.
The question now in my mind is, how did Milnor think of this algebra? I do not know the answer. Maybe he started with a round 4-sphere with its quaternionic Hopf fibration, and the elementary Morse function that consists of one of the coordinates in $\mathbb{R}^8$. You immediately get that there are two critical points (the north and south pole) and that they lie on the same Hopf fiber, since opposite points on a sphere always do. Apparently this Morse function fits together in a similar way for all of these 3-sphere bundles over the 4-sphere.