# Homology groups of a 2-sphere with q cross caps

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 $m(2a_i)$?

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