the space is obtained by attaching a circle $\mathbb S^1$ to the unit disk in $\mathbb R^2$ with boundary without the center by the attaching map $f(\cos t, \sin t) = (\cos 5t, \sin 5t)$?
I have learnt how to compute the fundamental group when attaching a 2-cell to a path-connected space, but $\mathbb S^1$ is not a 2-cell. So I don't know how to deal with it. Thank you.