computing fundamental group of $S^1 \times S^1$

Question

How to find $\pi_{1}(S^{1}\times S^{1})$ ?

I know $\pi_1(S^1)$ but ho to do this ?

$\pi_1(A\times B)=\pi_1(A)\times \pi_1(B).$ – Plutoro Nov 12 '15 at 17:24 — Plutoro, Nov 12 '15 at 17:24

score 0 · Accepted Answer · answered Nov 12 '15 at 17:24

0

Clearly it is isomorphic to $\pi_1(S^1)\times \pi_1(S^1)$ which is same as $\mathbb{Z} \times \mathbb{Z}$

answered Nov 12 '15 at 17:24

user289504

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......Clearly?! – Fly by Night Nov 12 '15 at 17:25
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I don't agree with the downvote (I upvoted) but I agree that this statement may not clear to someone who would ask this question. Anyway, the "explanation" is that a map $S^1\to S^1\times S^1$ is the same thing as a pair of maps $S^1\to S^1$ and $S^1\to S^1$, and similarly a homotopy, $S^1\times I\to S^1\times S^1$ is the same thing as a pair of homotopies $S^1\times I\to S^1$ and $S^1\times I\to S^1$. (If this isn't obvious, then you should probably convince yourself of this.) Therefore, $\pi_1(S^1\times S^1)\cong \pi_1(S^1)\times \pi_1(S^1)$. – Amitesh Datta Nov 12 '15 at 17:55
from next time you need to stay a bit more careful while writing your answer...you should always keep that in your mind that things might not be trivial for the person who asked a doubt, so you should try to be a bit more descriptive... It would be better if you edit your solution...btw welcome to mathstackexchange – Anubhav Mukherjee Nov 12 '15 at 21:36

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