Compute \(\pi_1(S^1 \vee S^1)\)
Compute \(\pi_1(S^1 \cross S^1)\)
Solution
1. The figure-eight satisfies\[\boxed{\pi_1(S^1\vee S^1)\cong F_2.}\]
Apply van Kampen to neighborhoods of the two circle summands whose intersection is contractible. The result is the free product \(\mathbb Z*\mathbb Z\).
2. The torus satisfies\[\boxed{\pi_1(S^1\times S^1)\cong\mathbb Z^2.}\]
Fundamental groups commute with products, so \(\pi_1(S^1\times S^1)\cong\pi_1(S^1)\times\pi_1(S^1)\cong\mathbb Z^2\).