I'm trying to compute the fundamental group of $\mathbb{RP}^2$ minus 2 points. I'm using the presentation $\langle a\mid a^2\rangle$. Meaning that I'm taking the disk and identifying the two sides.
I'm conjecturing that it is $\mathbb{Z}\times\mathbb{Z}$. I'm trying to use Seifert -Van Kampen but I can't get my open sets to work nicely. The reason why I think it is $\mathbb{Z}\times\mathbb{Z}$ is because clearly a loop around each hole would give you two distinct loops, say $a$ and $b$. But then it seems that $ab$ is homotopic to $ba$. This relation leads me to think that it might be $\mathbb{Z}\times\mathbb{Z}$.
Any hint would be appreciated.
Best Answer
Here is the standard way of tackling such problems:
Think of $\mathbb{RP}^2$ as as the disc $D^2$, with boundary identifiead in the oposite direction. Then after removing $2$ points the space retracts onto the space $X$. Now you can easily see that $X$ is nothing but $S^1\vee S^1$. So you have, $\pi_1(\mathbb{RP}^2\setminus\{\text{two points}\})\cong\pi_1(X)\cong\mathbb{Z}*\mathbb{Z}$