I’m reading the proof of Bott-Milnor-Kervaire 1,2,4,8 theorem and get stuck on the following process:
Why does the $w_n(\xi)$ not vanish?
Or is there another way to derive the 1,2,4,8 theorem from the fact that top Stiefel-Whitney class of vector bundles with rank $n$ over $S^n$ alway vanish except $n=1,2,4,8$?
Write $A$ for the division algebra, with basis $\{1, x_1, \cdots, x_{n-1}\}$.
There is a bundle with fiber $A$ given by the clutching construction using the left-multiplication map.
This bundle is constructed via the clutching construction, so one has a section which is just $1$ in the trivialization over the southern hemisphere. In the trivialization over the northern hemisphere, it takes the form $x \mapsto x$ on the boundary sphere. So one may construct a section over the north hemisphere which vanishes only at the North Pole; in radial coordinates $(r,x) \in [0,1] \times S^{n-1}$, the section takes the form $r^2x$. One may check that this is a transverse zero.
Because this bundle has a section with exactly one transverse 0, it has $w_n = 1$.