CW construction of Lens spaces Hatcher

I am working through Hatcher's book and I am having trouble while understanding the CW-complex structure of Lens spaces. It is on page 145. He proves it constructing it in an inductive process. I upload two pictures:

I think I understand all the previous ideas but I struggle with the text in yellow. I just don't see that. Why we have that? Are the $B_j^{2n-1}$ with $j \in \{1, \cdots ,m\}$ a cover of $S^{2n-3}$? I think there is something I have missed somewhere. Maybe I am not thinking about it adequately?

Some of my pictures so far are these (for the case $n=2$):

Maybe just explaining it with different words could help me to understand it. I am afraid I can not explain better.

Thanks in advance and any help would be appreciated. Also note there are two related questions: CW structure of lens space
CW construction of a generalized lens space from Hatcher

1 Answer

I think I got an answer to my own question. the $B_j^{2n-1}$ with $j \in \{1, \cdots ,m\}$ cover $S^{2n-3}$. That's easy to see when one write down the analytical formulas which are suggested here as a comment.