Understanding the transition map in the case of the proof that $S^3$ bundles over $S^4$

Question

Here is the part of the paper of "Rachel Mcenroe" on Milnor's construction of Exotic 7-spheres:

But I do not understand why the transition map is $\frac{1}{z}$ and it does not include any $\omega.$ Could anyone help me in understanding this ,please?

Answer 1

Regarding your first question, they have simply used a variable substitution $w=\frac{1}{z}$, from which one computes $$\phi_2 \circ \phi_1^{-1}(z) = \phi_2 (\phi_1^{-1}(z)) = \phi_2([z;1]) = \phi_2([\frac{1}{w};1]) = \phi_1([1;w]) = w = \frac{1}{z} $$