Fraleigh(7ed) Section43 Exercise9
The below is the full solution. But I can't understand the red underlined parts.
1) Why the piece that contains the i-th crosscap have boundary $z_i-2a_i$? I don't know how to compute this boundary. Isn't it just the $z_i$? How the $-2a_i$ parts come from?
2) I think that this part will be similar to the (1). How to compute $\partial_2 (c)$?
3) I thought that any $2a_i$ is a 1-boundary of the i-th crosscap. Why the $2a_i$ are not boundaries, but only the sum of all $m(2a_i)$?