Fundamental group of the quotient space

Can anyone help me with computing this fundamental group...

Namely, I have been given a standard (closed) disk and a sphere resp., i.e. $\mathbb D^2$, $\mathbb S^1$ resp. in a complex plain $\mathbb C$.

Now, let $X$ be $X:=\mathbb D^2/\sim$ where $\sim$ stands as a relation such that $x\sim ix$ for each $x\in\mathbb S^1$.

How can I then compute fundamental group $\pi_1(X,0)$?

I have been trying to use van Kampen theorem somehow, but didn't get much of the result.

• Yeah, Van Kampen is the right tool. One set is a neighborhood of $0$, another the complement of zero: the image of the generator of $\pi_1$ of the intersection in one set is trivial, in the other set it's 4 times the generator... – Peter Franek Feb 12 '17 at 23:01
• In this case, it is easier to apply the theorem formally than to have a good intuition about this space. It is not a manifold or anything very nice. It may help to represent it as a square with all four sides identified, if you like it more. – Peter Franek Feb 12 '17 at 23:10
• Okay. If I take it only formally, for a little nbh of $0$, say $U$, I can take, say, small disk whose fundamental group is trivial. But what about intersection and the other open set, say $V$... I am not sure if I get this generator hint above.. – edward_scissorhands Feb 12 '17 at 23:12
• If $\alpha$ is the generator of $\pi_1(U\cap V)$ and $\beta$ the generator of $\pi_1(V)$ what is the image of $\alpha$ in $\pi_1(V)$? – Peter Franek Feb 12 '17 at 23:18