Calculate the fundamental group of $S^1/\mathbb Z_n$ ,where $\mathbb Z_n$ acts naturally on $S^1$ by rotations of $2\pi /n$
The origin of this problem is the following unclear solution of another problem:
I restate the question in picture here:
PROBLEM Compute $\pi_1 (X)$, where $X$ is the quotient space of a torus $S^1\times S^1$ obtained by identifying points on the circle $S^1 \times\{y_0\}$ that differ by rotations of the circle by $\frac{2\pi}{n}$ for $y_0$ a base point in the second $S^1$ factor and $n\ge2$ a fixed integer
My question is how to give a rigor and systematical answer, not just some intuitive or geometrical answers. What puzzled me is that in the above picture, it claims that the fundamental group of $S^1/\mathbb Z_n$ is $\mathbb Z_n$ . However, I guess $S^1/\mathbb Z_n$ seems to be homeomorphic to $S^1$; hence, the fundamental group should be $\mathbb Z$.
I think here we cannot use Van-Kampen theorem and we must back to the definition. Please help!