[Math] The fundamental group of the connected sum of two copies of the real projective plane

algebraic-topologyfundamental-groupsgeneral-topology

How do I compute the fundamental group of the connected sum $X \mathop{\#} X$, where $X$ denotes the real projective plane?

I'd like to use Van Kampen's theorem, but I have trouble visualizing what this space looks like. Could anyone provide hints?

Best Answer

The real projective plane $\mathbb{RP}^2$ can be represented as the following identification bi-gon:

The fundamental polygon of the real projective plane

By taking two of these bigons, cutting each at a vertex, and gluing them together again, we obtain the identification polygon for $\mathbb{RP}^2 \# \mathbb{RP}^2$:*

The fundamental polygon of the connected sum of two real projective planes

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 $P$.

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