1
$\begingroup$

enter image description here

Consider the covering space of $S^1 \vee S^1$ in $(1)$. Then distinct loops in $(1)$ are represented by $\langle a, b^2, bab^{-1} \rangle$.

Thus elements of the fundamental group are words generated by these distinct loops.

This fundamental group will map to a subgroup $H$ of $\pi_1( S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z}$ under the covering map.

With this covering map, I think, $a \mapsto a$, $bab^{-1} \mapsto a$ and $b^2 \mapsto b^2$.

I can't quite put this information together to determine $H$. My guess would be $\langle a, b^2 \rangle$ but I am not quite sure if this is correct.

Any help is appreciated!

$\endgroup$
1
$\begingroup$

1)is $S^1\vee S^1\vee S^1$ its fundamental group is $Z*Z*Z$ and it is the subgroup of $\pi_1(S^1\vee S^1)$ generated by $a,b^2,bab^{-1}$.

$\endgroup$

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.