I am reading page 144 from hatcher's book about lens space, but I am really confused about the CW-structure described there.

He first takes one unit circle from one factor of $\mathbb C^n$, then divide it into m parts. Then he join jth vertex to $\mathbb S^{2n-3}$ in $\mathbb C^{n-1}$ by arcs of great circles in $S^{2n-1}$. Then he says we get a ball $B_j^{2n-2}$ bounded by $S^{2n-3}$. I am really confused about how this ball is got. For a $S^{2n-3}$, the interior is already a ball, so how do we get a ball by attaching arc to this sphere?

By the way, does any one know other material to learn the CW-structure of lens space?




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