# Fundamental group of the quotient space of the disk obtained by identifying points on the boundary that are 120 degree aparts

Let $X$ be the quotient space of the disk, $\{(x,y)\in \mathbb R^{2} \ | \ x^{2}+y^{2}\leq 1 \}$, obtained by identifying points on the boundary that are $120$ degrees apart. How can we find the fundamental group of $X$ ?

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is it a homework? It's a good manner to tell us what you tried. – user8268 Jun 3 '11 at 20:16
No, it is a past prelim question, and I think I need to use the Van Kampen theorem but I don't know how – hebele Jun 3 '11 at 20:22
It's cyclic of order 3. You can use Van Kampen's theorem directly or you can just put a cell structure on the quotient. – Jim Belk Jun 3 '11 at 20:28
I suppose you learned how to compute $\pi_1$ of a cell complex. In this case th 1-skeleton $X^1=S^1$ and $X$ is obtained by gluing a disk with the boundary circle going 3x around $X^1$. Is it helpful enough? – user8268 Jun 3 '11 at 20:33
@hebele - if you have done the projective plane before (identify antipodal points - i.e. 180 degrees), you should able to do this! (and you should be able to guess the answer straight away) – Juan S Jun 3 '11 at 22:56