In section 12 of Milnor-Stasheff's characteristic classes, they mentioned an obstruction class for the associated Stiefel manifold bundle $V_k(\xi)$ over $B$ as follows:
Definition: The associated Stiefel manifold bundle $V_k(\xi)$ over $B$ for an $n$-plane bundle $\xi$ over $B$ is defined as the bundle obtained from gluing $V_k(F_x)$, where $x\in B$, the base manifold and $F_x$ is a fibre of $\xi$ over $x$.
Existence of cross section of $V_k(\xi)$ over the $(n-k)$-skeleton of $B$ is guaranteed, but not for the cross section over the $(n-k+1)$-skeleton. There exists a cross section over the $(n-k+1)$-skeleton of $B$ iff some obstruction class in $H^{n-k+1}(B;\{\pi_{n-k}V_k(F)\})$ vanishes.
Definition: We use the notation $\mathfrak{o}_{n-k+1}(\xi)\in H^{n-k+1}(B;\{\pi_{n-k}V_k(F)\})$ to denote such obstruction class.
Now I am confused with the argument here in the proof of theorem 12.1:
In their construction there I can see why it gives a cross section over the $(n-1)$-skeleton, but why the obstruction cocycle "assign" the $n$-cell to a generator of the homotopy group in the very last line?
The argument to me there is not clear for me.
It will be great if someone can give a geometric interpretation of such obstruction class.
Thanks in advance for your attention. Let me know if something is not clear in my question!