# The fundamental group of some wedge sum

I was wondering how one can compute the fundamental group of the wedge sum of a sphere and 2 circles , i know the fundamental group is Z*Z ,and that the fundamental group of a wedge sum is the free product of each fundamental group but is it possible to get it by Van Kampen ? I couldn't do it .Any help is appreciated .

• Are you able to compute the fundamental group of $S^1 \vee S^1$ with Van Kampen's theorem? It's really the same idea... – Najib Idrissi Apr 19 '15 at 13:35
• yes i am able to do it for 2 circles , but the thing is that i can't find this division of the space that gives me the intersection of the three subspaces not empty , in the wedge sum of 2 circles it's easier,but here how can I get an intersection between the three subspaces ?@NajibIdrissi – Butterfly Apr 19 '15 at 13:39
• Apply Van Kampen's theorem twice: once to get $\pi_1(S^2 \vee S^1 \vee S^1) \cong \pi_1(S^2) * \pi_1(S^1 \vee S^1)$, then once again to get $\pi_1(S^1 \vee S^1) \cong \pi_1(S^1) * \pi_1(S^1)$. See hunter's answer. – Najib Idrissi Apr 19 '15 at 13:41
• Yes, that's the idea. – Najib Idrissi Apr 19 '15 at 13:42
• Sorry i deleted my question because you've already answered it .Thank you @NajibIdrissi – Butterfly Apr 19 '15 at 13:43

The two techniques you mentioned in your question: "the fundamental group of a wedge sum is the free product of each fundamental group" and "getting it by Van Kampen" are the same. Let $X$ and $Y$ be any nice spaces (we'll specify what nice means below) and let $S$ be their wedge sum at a common point, which we'll call $p$.

Then the copy of $X$ in $S$ admits an open neighborhood $U_X$ which is the union of the copy of $X$ and a small open neighborhood of $p$ in $Y$ which deformation retacts to $p$ (so one reasonable definition of "nice" would be "each point has a contractible neighborhood," although this can be relaxed -- but at least it certainly applies in your example). Define $U_X$ similarly. Since $U_X \cap U_Y$ is contractible, you recover "the fundamental group of a wedge sum is the free product of each fundamental group" from Van Kampen's theorem.

• I understand the idea now ,Thank you for your answer . – Butterfly Apr 19 '15 at 13:44