if we have the wedge product of the real projective plane $P^2$ V $P^2$

Then how would i use Seifert Van Kampens theorem to compute the fundamental group $\pi_1$($P^2$ V $P^2$ ) ?

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

  • $\begingroup$ how do you define this wedge product? $\endgroup$ – janmarqz Nov 30 '14 at 19:21

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

  • $\begingroup$ thank you, as the fundamental group of $P^2$ = $\mathbb{Z}_2$ could you elaborate as to how you got to this stage? $\endgroup$ – user123 Nov 30 '14 at 19:43
  • $\begingroup$ Seifert-van Kampen says that if $X=A\cup B$, but $A\cap B$ is a point, then $\pi_(X)=\pi_1(A)*\pi_1(B)$ i.e. their free product. $\endgroup$ – janmarqz Nov 30 '14 at 19:47
  • $\begingroup$ ah of course, from the definition of the wedge product there exists only a single point of intersection, thanks! $\endgroup$ – user123 Nov 30 '14 at 19:57
  • $\begingroup$ you are thru man $\endgroup$ – janmarqz Nov 30 '14 at 20:00

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.

  • $\begingroup$ thanks for the answer! im guessing a good choice of base point would be the point where the two copies join? $\endgroup$ – user123 Nov 30 '14 at 19:31
  • $\begingroup$ @user123 So long as you take advantage of it! $\endgroup$ – Benjamin Nov 30 '14 at 19:35
  • $\begingroup$ @user123 You should also only accept my answer if it has completely satisfied you. $\endgroup$ – Benjamin Nov 30 '14 at 19:36
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    $\begingroup$ or wait a little more to see if there are more answers XD $\endgroup$ – janmarqz Nov 30 '14 at 19:37
  • $\begingroup$ @Benjamin, thanks again! and sorry its the first day of me using this forum i need to get used to the rules! $\endgroup$ – user123 Nov 30 '14 at 19:44

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