2
$\begingroup$

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 .

$\endgroup$
  • 1
    $\begingroup$ 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... $\endgroup$ – Najib Idrissi Apr 19 '15 at 13:35
  • $\begingroup$ 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 $\endgroup$ – Butterfly Apr 19 '15 at 13:39
  • 1
    $\begingroup$ 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. $\endgroup$ – Najib Idrissi Apr 19 '15 at 13:41
  • $\begingroup$ Yes, that's the idea. $\endgroup$ – Najib Idrissi Apr 19 '15 at 13:42
  • $\begingroup$ Sorry i deleted my question because you've already answered it .Thank you @NajibIdrissi $\endgroup$ – Butterfly Apr 19 '15 at 13:43
1
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.