I am having difficulty understanding the following remark made by Hatcher on page 50 of Algebraic Topology:
My understanding of this paragraph is as follows: For a basepoint $s_0\in S^1$ the attaching map $\varphi_{\alpha}$ determines a loop based at $\varphi_{\alpha}(s_0)$ in $Y$, namely the boundary of the image of $e^2 _{\alpha}$. For some path $\gamma_{\alpha}$ from $x_0$ to $\varphi_{\alpha}(s_0)$ one can use the change of base isomorphism, one sees that the loops $\varphi_{\alpha}(s_0)$ can be associated with loops all based at the same point $x_0 \in X$. This is as far as I can follow. I don't understand why although the loop $\gamma_{\alpha}\varphi_{\alpha}(s_0)\overline{\gamma_{\alpha}}$ will be nullhomotopic after attaching $e^2_{\alpha}$, even though it may not be before.
Why is this? Also, are there examples which can elucidate why this is true?