I'm trying to find the fundamental group of space $X$ obtained from an annulus $\{p \in \mathbb R^2 : 1 \leq |p| \leq2 \}$ by identifying the point $(x,y)$ on the inner circle of radius $1$ with the point $(-2x,-2y)$ on the outer circle of radius $2$.
I think that we can use CW-complexes do that, but I'm not sure how to do it in this case. Moreover, I think I can apply van Kampen here too by taking the sets $A= X \smallsetminus \{\text{inner circle}\}$ and $B= X \smallsetminus \{\text{outer circle}\}$. Is that correct?
Thanks...