Suppose $X$ is obtained by gluing two tori at a single point and let $r:\sum_2\to X$ be the retraction given by collapsing a circle around the middle of $\sum_2$ (surface of genus $2$) to a single point $x_0$.
I am trying to describe the induced homomorphism $r_{*}:\pi_1(\sum_2,x_0)\to \pi_1 (X,x_0)$.
I can see that $\pi_1(X,x_0)=\mathbb{Z}^{2} * \mathbb{Z}^{2}$. But I am not sure how I can make use of this information to describe the induced homomorphism $r_*$. Any hint/help will be very useful.
Thanks.
Well, $\pi_1(\Sigma_2, x_0)$ has the "standard" presentation $\langle a_1, b_1, a_2, b_2 : [a_1,b_1][a_2,b_2]\rangle$. Draw a picture of $\Sigma_2$ and these generators of its fundamental group!
$\pi_1(X,x_0)$ has the "standard" presentation $\langle x_1, y_1, x_2, y_2 : [x_1,y_1], [x_2,y_2] \rangle$. Draw a picture of $X$ and these generators of its fundamental group!
Now you can look at your pictures to determine: where does $r_*$ send $a_1, b_1, a_2, b_2$?