# The fundamental group of a wedge sum applying Van Kampen 's theorem

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?

• 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. May 14 '17 at 17:58
• @MichaelAlbanese True but with a simply connected extension you get a free product rather than an amalgamated product, which is a bit nicer. 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.
• It's the free product of $\pi_1(S^1)$ and $\pi_1(S^2)$. Is that $\Bbb Z$? May 14 '17 at 18:10
• @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? May 14 '17 at 18:35