I am fairly new to algebraic topology so please bare with me if this seems simple
I am trying to find the fundamental group of the unit disk with the identification on the boundary z = (cos(θ), sin(θ)) being mapped to (cos(θ+2π/n), sin(θ+2π/n)).
For n=1 it is just the disc so the fundamental group is trivial (since the disk is convex).
Therefore, I was trying to solve it for n=2 to begin. With n=2 each point on the boundary is being identified to its antipodal point.
I was trying figure out a way to use van kampen's theorem.
As I know it, van kampen's theorem states that if X = A∪B (A and B open) and A∩B path-connected, then π1(X) = [π1(A) * π1(B)] / π1(A∩B)
(where * is the free product and you quotient out by the fundamental group of the intersection)
I think that there probably is a way by letting A equal the interior of the disk, but I am not sure what to make B since I am even having trouble seeing what the open sets in the quotient space are.
Thanks for the help
You can take $B$ as the image of the annulus $\{ z \ | r< |z| \le 1\}$. This is an open subset in the quotient space, as the image of a saturated open subset ( saturated meaning: any point equivalent to a point in that subset is still in the subset).
(As a side notice, for a surjective quotient map of topological spaces the open subsets in the target space are the images of saturated open subsets in the source).
To apply the Seifert-van Kampen theorem, it's best to think in terms of diagrams and universal objects. $\pi_1(A\cup B)$ is a direct sum with identifications, the way $A\cup B$ is a direct sum of $A$ and $B$ with identifications along $A\cup B$. So one has a category of nice commutative diagrams and $\pi_1(A\cup B)$ is (more or less) an initial object. But basically it's $\mathbb{Z}/n$.