I have a question. I am trying to calculate $\pi _{1}(X)$ where $X$ is a polygon as you see in the picture such that $p_{1}$ and $p_{2}$ two punctures by identifying edges.
Here is my idea. I think, if I identify edges I will have $2-$Sphere with two punctured points. One point punctured sphere homotopy equivalent to $\mathbb{R}^{2}$, then by using the second one, it is homotopy equivalent to $S^{1}$. And finally, I will have $\pi _{1}(X)=\mathbb{Z}$. Am I right?
Thank you for your help.