wedge product of projective planes

Question

If we have the wedge product of the real projective plane $\mathbb{RP}^2 \vee \mathbb{RP}^2$.

Then how would i use Seifert Van Kampens theorem to compute the fundamental group $\pi_1$($\mathbb{RP}^2 \vee \mathbb{RP}^2$ ) ?

I'm some what confused on Van Kampens theorem especially when applying it to the real projective plane

any help on this would be greatly appreciated! thank you

Answer 1

The Seifert-van Kampen will give the answer: $\pi_1(P^2\vee P^2)=\mathbb{Z}_2*\mathbb{Z}_2$.

Answer 2

Let $x_0$ be the point of $P^2 \vee P^2$ where we glued the two copies of $P^2$ together.

Recall that to apply Van Kampen theorem (and obtain an isomorphism), we need to write $P^2 \vee P^2$ as the union of path-connected open sets, each containing $x_0$, such that the intersection of any three is path-connected. We should aim to write $P^2 \vee P^2$ using only two path-connected open sets, as that will make our job much easier.

My hint would be to take advantage of the fact that you have two copies of $P^2$, which you know is path-connected (right?) and whose fundamental group you know (right?), after all you should want to write the fundamental group of $P^2 \vee P^2$ (what should your basepoint be?) in terms of some well-known group.