$\pi_1(S^1 \times S^1)$

Solution

1. For path-connected spaces, fundamental groups commute with products:\[\pi_1(X\times Y,(x_0,y_0))\cong\pi_1(X,x_0)\times\pi_1(Y,y_0).\]

A loop in a product is exactly a pair of loops, and homotopies are likewise coordinatewise.

2. Since $\pi_1(S^1)\cong\mathbb Z$,\[\boxed{\pi_1(S^1\times S^1)\cong\mathbb Z\times\mathbb Z.}\]

Apply 1 to the two circle factors.

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