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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

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  • 1
    $\begingroup$ Hint: $\Bbb{R}P^2\#\Bbb{R}P^2\#\Bbb{R}P^2$ is homeomorphic to $T^2\#\Bbb{R}P^2$. $\endgroup$ – Santana Afton May 13 at 7:14
  • $\begingroup$ Use the nice CW structure of $n\#\mathbb{RP}^2$, or compute its (co)homology and use the classification of closed surfaces. $\endgroup$ – anomaly May 13 at 12:23

Assuming that it is a connected sum of n copies of the projective plane, yes, that is a correct presentation for the fundamental group. One thing to keep in mind is that just because you know a presentation of a group, you don't necessarily 'know the group'. As an interesting exercise, can you tell if these groups are different for different values of n?

BTW, you don't need to know about CW complexes or cohomology or the classification of surfaces to do this problem.


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