Crushing a central separating circle in $\Sigma_2$ to a point
So presentation of $\Sigma_2$ is $<x_1,y_1,x_2,y_2:[x_1,y_1][x_2,y_2]>$.
I understand that crushing this separating circle to a point will turn the space into a wedge of two tori and so the fundamental group will be $Z \oplus Z*Z \oplus Z$, but I was hoping somebody could explain to me, in terms of the group presentation, how this comes about. How does collapsing the separating circle destroy change the presentation to
$\Sigma_2 = < x_1,y_1,x_2,y_2: [x_1,y_1]=1, [x_2,y_2]=1>$?
Crushing the central circle to a point is of course homotopy equivalent to gluing a disk with that circle as boundary. Since that circle is $[x_1,y_1]$, you have $$ \langle x_1,y_1,x_2,y_2\mid [x_1,y_1][x_2,y_2],[x_1,y_1]\rangle =\langle x_1,y_1,x_2,y_2\mid [x_1,y_1],[x_2,y_2]\rangle $$