Just wanted to confirm if my proof is correct and complete, trying to learn Van-Kampen Theorem.

Question: Find the fundamental group of two copies of $S^2$ attached at a point .

Proof: We claim that $\pi_1(X)$ is trivial.

Let the two copies of $S^2$ be U and V. Then $X=U \cup V$.

$U \cap V = {p}$, where $p$ is the point of attachment, and hence it is path connected. We know that $\pi_1(U)$ and $\pi_1(V)$ are trivial.

So by Van-Kampen, the fundamental group of $X$ is generated by fundamental groups of $U$ and $V$, but since they are both trivial, $\pi_1(X)$ is also trivial.

  • 12
    $\begingroup$ Careful, the actual statement of Van-Kampen's theorem requires that $U$ and $V$ be open. (The actual idea of your proof is right, you just need a little 'adjustment'). $\endgroup$
    – Dan Rust
    Commented Apr 24, 2013 at 20:52
  • $\begingroup$ Thanks Daniel. I guess U and V have to be sets other than the spheres themselves. $\endgroup$ Commented Apr 25, 2013 at 20:50
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    $\begingroup$ It's an easy fix. What's the most natural open subset of $S^2\vee S^2$ you can think of, which isn't the whole space, and contains one entire sphere? $\endgroup$
    – Dan Rust
    Commented Apr 25, 2013 at 21:41


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