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Let $X=\{(x,y) \in RP^n\times RP^n;x=x_0, or, y=x_0\}$ where $x_0$ is some fixed point of $RP^n$. In other words, $X$ is two copies of $RP^n$ with one point $x_0$ in common. find $\pi_1( X,x_0)$?

I think the fundamental group of $RP^n$ is $Z_2$, so the join $\pi_1( RP^n ∨ RP^n)$ should be $Z_2*Z_2$. but can I confirm this use the Seifert–van Kampen theorem. I am not sure what will the $U$ and $V$ be as in the theorem, so that I can write down a formal proof. Thanks in advanced, any help is appreciated!

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Take $U$ to be $\mathbf{RP}^n_1$ union a small neighborhood of $x_0$ in $\mathbf{RP}^n_2$, and $V$ to be $\mathbf{RP}^n_2$ union a small neighborhood of $x_0$ in $\mathbf{RP}^n_1$. We can take the small open neighborhoods in question to be contractible. Then $U \cap V$ is also contractible.

Exercise: in what greater generality does this argument apply?

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  • $\begingroup$ Thanks a lot for answering! I think the generality is the $n$ copies of $RP^n$ with one point $x_0$ in common has the fundamental group as the free product of $n$ copies of $Z_2$ ? $\endgroup$ – user138017 Apr 9 '15 at 8:09

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