I need to compute the fundamental group of the wedge sum $\mathbb{S}^1 \vee \mathbb{S}^2$. I want to find two open sets $U$ and $V$ such that their intersection is a simply connected space so I could apply Seifert Van Kampen theorem. Any ideas please?

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    $\begingroup$ Note, you don't need the intersection of the two open sets to be simply connected to apply van Kampen's theorem, you only need it to be path-connected. $\endgroup$ May 14 '17 at 17:58
  • $\begingroup$ @MichaelAlbanese True but with a simply connected extension you get a free product rather than an amalgamated product, which is a bit nicer. $\endgroup$ May 14 '17 at 18:11

Just take a point from the $S^1$ away from $X$ to give $U$ (not the basepoint of course) and just take a point from the $S^2$ away from $X$ to give $V$ (not the basepoint of course). Then $U\cap V$ is a wedge sum of $\Bbb R^1$ and $\Bbb R^2$ which is easily shrunk down to the basepoint.

  • $\begingroup$ The fundamental group i am looking for is Z right? $\endgroup$
    – user391120
    May 14 '17 at 18:05
  • $\begingroup$ It's the free product of $\pi_1(S^1)$ and $\pi_1(S^2)$. Is that $\Bbb Z$? $\endgroup$ May 14 '17 at 18:10
  • $\begingroup$ Of course not but i am asking because in this link eclass.uoa.gr/modules/document/file.php/MATH212/… page 46 claims this $\endgroup$
    – user391120
    May 14 '17 at 18:29
  • $\begingroup$ @user391120 That's all Greek to me! (sorry, couldn't resist!). Do you know what $\pi_1(S^1)$ and $\pi_1(S^2)$ are? $\endgroup$ May 14 '17 at 18:35
  • $\begingroup$ the fundamental group of first is the set of the integers and for the second {0} or not? I meant the second shape in page 46 .I am sorry for all this trouble and many thanks for your help $\endgroup$
    – user391120
    May 14 '17 at 18:40

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