this is a topology question, it's like this:
Find the fundamental group of
1) (Z × Z) ∗ Z.
2) (Z ∗ Z) × Z.
3)Z ∗ · · · ∗ Z where there are n copies
Here, Z stands for the integer set, * is the free product.
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My thoughts:
I know $\pi( S^1)$ = Z，so intuitively, I have
1) $\pi_1 ((S^1 \times S^1) \vee S^1) $
2) $\pi_1 ((S^1 \vee S^1) \times S^1) $
3) $\pi_1 ((S^1 \times ... \times S^1) \vee S^1) $ where there are n copies of $S^1$ being wedge summed.
However, I’m not sure how to prove I got it right, like how to use theorems or math proofs to show it, or is it not needed in this case?
Please tell me whether i’m correct or not, and help me with how to derive an answer.
Thank you in advance for the help!
The first two are correct. $\pi_1(S^1 \times S^1)=\mathbb Z \times \mathbb Z$ follows from the universal property of products in topology (so homotopy groups "split" over cartesian product.
The free product follows from local contractibility and Van Kampen's theorem so that $\pi_1(X \vee S^1)=\pi_1(X) *\mathbb Z $ by choosing appropriate neighborhoods.
The third one is incorrect. This will be $\mathbb Z^{\oplus n} *\mathbb Z$ rather than the free product of $\mathbb Z$'s. See roses and try van kampen's theorem to prove the calculation of their fundamental groups.
if you want a more general approach that does not consist of building up new groups from ones you know already (for example more complicated groups) you should see the presentation complex which is a space with one vertex, one $1$-cell for each generator of your group and a $2$-cell to impose relations on the group. This is related to realizing a group as a quotient by a free group.