# What is the fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ [duplicate]

This question already has an answer here:

I want to know about fundamental group of $$RP^{2}$$ # $$\cdots$$ # $$RP^{2}$$ by Seifert-Van Kampen theorem.

In my guessing, that is $$\langle a_1, a_2 ,... a_n | a_1^{2}a_2^{2}\cdots a_n^{2}=1\rangle$$.

Is this correct?

## marked as duplicate by user10354138, Community♦May 28 at 6:03

• Hint: $\Bbb{R}P^2\#\Bbb{R}P^2\#\Bbb{R}P^2$ is homeomorphic to $T^2\#\Bbb{R}P^2$. – Santana Afton May 13 at 7:14
• Use the nice CW structure of $n\#\mathbb{RP}^2$, or compute its (co)homology and use the classification of closed surfaces. – anomaly May 13 at 12:23