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I know that a torus the product of circles. But what the fundamental group of the product of a 3-sphere and a circle? ie $\pi_1((S^3 \times S^1), (1,1))$?


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The fundamental group of a product is the product of the fundamental groups. Since $\pi_1(S^3)=\{0\}$, that means that the fundamental group is just $\pi_1(S^1)=\mathbb Z$ – Thomas Andrews Nov 30 '11 at 14:41

Given the projection maps $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$, the map $$\pi_1(X \times Y)\to \pi_1(X)\times \pi_1(Y);\;\gamma\mapsto (p_X\circ \gamma\;,\;p_Y\circ \gamma)$$ is an isomorphism

Now a sphere $S^n$ is $(n-1)$-connected, meaning that $\pi_i(S^n)=0$ for all $i=0..n-1$. So $\pi_1(S^3)=0$. On the other hand $\pi_1(S^1)=\mathbb Z$ this gives that $\pi_1(S^3\times S^1)=\mathbb Z$.

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