I'm struggling to find the fundamental group of the following surface.
Let $\tilde{X}$ be the space constructed by taking the $3$-ball and cutting out an open solid torus from inside. Let $p$ and $q$ be points on the two connected components of the boundary of $\tilde{X}.$ The space $X$ is obtained by gluing $p$ and $q,$ i.e., $X=\tilde{X}/(p\sim q).$ Find $\pi_1(X).$
Linked above is the image given. I first tried to think about how I would find $\pi_1(\tilde{X}),$ i.e., before gluing $p$ and $q,$ and I believe that $\pi_1(\tilde{X})\cong\mathbb{Z}.$ However, I'm not sure how to think about the fundamental group when the two points are glued. My instinct is that this doesn't change the fundamental group, but I'm not sure how to show this (by SvK, for example).
How should I think about this? Thank you in advance!