algebraic-topologyfundamental-groupsprojective-space
I want to compute the fundamental group of $\mathbb R \mathbb P_2 \backslash \{A,B\}$ (where $A$,$B$ are points) but I am stuck. I tried to use van Kampen's theorem bit I can't find two good subset (I tried $A=\mathbb R \mathbb P_2 \backslash \{A,B,p\}$ (where $p$ is anorher point) and $B=(\mathbb R \mathbb P_2 \backslash \{A,B\})^\circ$, but I can't manage to compute their fundamental groups).
Any hints? Am I on the right way with using Van Kampen, or should I try something else?
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, $\pi_1(\mathbb{RP}^2\setminus\{\text{two points}\})\cong\pi_1(X)\cong\mathbb{Z}*\mathbb{Z}$
The real projective plane $\mathbb{RP}^2$ can be represented as the following identification bi-gon:
By taking two of these bigons, cutting each at a vertex, and gluing them together again, we obtain the identification polygon for $\mathbb{RP}^2 \# \mathbb{RP}^2$:*
Now, let $P$ be an interior point of the identification polygon. Then, we can apply the Seifert-van Kampen Theorem to $\mathbb{RP}^2 \# \mathbb{RP}^2 = \mathbb{RP}^2 \# \mathbb{RP}^2 \setminus \{P\} \cup D$, where $D$ is a small disk containing $P$. **
Can you take it from here?
* : Note that $\mathbb{RP}^2 \# \mathbb{RP}^2$ is in fact homeomorphic to the Klein bottle. This can be seen by cutting and regluing the identification polygon.
** : This is a standard trick for finding fundamental groups of compact surfaces. We first realize the surface as an identification polygon, and then apply the Seifert-van Kampen Theorem to the polygon with a point $P$.removed and a small disk around $P$.
Best Answer
Well, you can think of $\mathbb{R}\text{P}_{2}$ as the upper hemisphere of a $2$-sphere with antipodal maps on the boundary identified. (See here: Real Projective Plane is Same as Identifying Antipodal Boundary Points of The $2$-Disc.). Then, choose any two points on the upper hemisphere, and get rid of those. You can continually deform the resulting space into the boundary of $\mathbb{R}\text{P}_{2}$ along with an arc joining any two points on the boundary. This space is the same the wedge sum of two copies of $S_{1}$, and by Van - Kampen's theorem, the fundamental group of this space is the free product of two copies of $\mathbb{Z}$, i.e. $\mathbb{Z} * \mathbb{Z}$.