Fundamental group of the torus $T^2=S^1\times S^1$

First, use the product theorem for fundamental groups: \[ \pi_1(S^1\times S^1,(1,1)) \cong \pi_1(S^1,1)\times\pi_1(S^1,1) \cong \mathbb Z\times\mathbb Z. \]

Second, use the standard square CW structure on the torus. After identifying opposite edges, there is one \(0\)-cell, two \(1\)-cells \(a,b\), and one \(2\)-cell whose attaching map follows the boundary word \[ aba^{-1}b^{-1}. \] Van Kampen therefore gives \[ \pi_1(T^2)\cong\langle a,b\mid aba^{-1}b^{-1}=1\rangle. \] The single relation says exactly that \(a\) and \(b\) commute, so this group is \[ \mathbb Z^2. \] Thus both methods give \[ \boxed{\pi_1(T^2)\cong\mathbb Z\oplus\mathbb Z}. \]