In lectures, we have shown that the fundamental group of $\mathbb{R^n}$ is $\{0\}$ and $S^1$ is isomorphic to $\mathbb{Z}$. I was wondering how one might go about computing the fundamental groups of more complicated spaces, such as $S^1 \times D^2$ and $S^1 \times S^1$. Thank you in advance!
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2The fundamental group functor preserves products. – Angina Seng Nov 05 '17 at 20:05
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Read some more questions at MSE, e.g. this one for more interesting examples, such as the figure eight space: $\pi_1( S^1 \vee S^1) =\mathbb Z \ast \mathbb Z$. Direct products are not too interesting. – Dietrich Burde Nov 05 '17 at 20:17
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Computations of fundamental group can be quite difficult. Some techniques include:
The seifert-van Kampen Theorem
Long Exact Sequence of Fibration
Passing to the universal cover.
In your cases, these all follow from the fact that $\pi_1(X \times Y)\cong \pi_1(X) \times \pi_1(Y)$ for path connected spaces. This follows essentially by checking a bijection by using projection maps, and that a continuous map in $X \times Y$ is continuous if and only if it is continuous in each factor.
Andres Mejia
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It's probably important to point out that $X$ and $Y$ have to be path connected for $\pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y)$ to hold. – Exit path Nov 05 '17 at 21:48
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Other techniques: homotopy equivalence ($S^1\times D^2$ deformation retracts onto $S^1$) and CW decomposition ($S^1\times S^1$ has a CW structure of a point, two circles, and disc; the circles generate $\pi_1$ and the discs give relations). – Kyle Miller Nov 06 '17 at 06:43