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Does anyone know a way of computing the fundamental group of the connected sum of $n$ copies of $\mathbb{R}P^n$?

Any help will be appreciated. Thank you!

Susan
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    Seifert Van-Kampen theorem? – Ryan Budney Apr 10 '13 at 05:47
  • i have no clue how to compute the general case though. – Susan Apr 10 '13 at 05:55
  • Instead of thinking of it as $\mathbb RP^n$ connect sum itself, perhaps you should think of it as $S^n$ connect-sum many $\mathbb RP^n$ ? This way all the discs are in one common manifold and it will be easier to identify the pattern. – Ryan Budney Apr 10 '13 at 05:56

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For $n=1$, it is $\pi_1(S^1) \cong \mathbb{Z}$.

For $n=2$, we have $\mathbb{R}P^2 \# \mathbb{R}P^2 \simeq \textit{Klein Bottle}$, so $\pi_1(\mathbb{R}P^2 \# \mathbb{R}P^2) \cong \mathbb{Z} \rtimes \mathbb{Z}$.

For $n \geq 3$, according to Fundamental group of the connected sum of manifolds the fundamental group of a connected sum of $\geq 3$-dimensional manifolds is the free product of the fundamental groups, so $\pi_1(\#_{i=1}^n \mathbb{R}P^n) \cong *_{i=1}^n\mathbb{Z}/2$.

Alex Provost
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