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I'm thinking about the fundamental group of a circle with some points identified. I mean let $r:\mathbb S^1\to \mathbb S^1$ be a quotient map mapping the point of the circle $(cos \theta, sin \theta )$ to $(cos(\theta+2\pi /n),sin (\theta+2\pi /n))$. Form a quotient space identifying $x$ to $r(x), r^2(x), \ldots,r^{n-1}(x)$.

I need help to find this fundamental group.

Thanks

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    You probably mean $(\cos(\theta), \sin(\theta)) \mapsto (\cos(\theta + \frac{2\pi}{n}), \sin(\theta + \frac{2\pi}{n}))$. Anyway, can you identify the quotient space itself? Say, when $n = 2$?2012-11-25
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    @levap yes of course thank you, I'm editing right now2012-11-25
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    @levap Maybe we have a homeomorphism between a bouquet of n circles and this space, am I correct?2012-11-25
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    Not really. In a bouquet of $n$ circles there is a point at which you "wedge" the circles. Which point would it be in the quotient? When $n = 2$, a fundamental domain for the action is the half circle $(\cos(\theta), \sin(\theta))$ where $0 \leq \theta < \pi$. That is, each equivalence class in the quotient has two members and you can always choose a member in the equivalence class to lie in the half circle. What is the relation between $(\cos(0),\sin(0))$ and $(\cos(\pi), \sin(\pi))$ in the quotient?2012-11-25

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