(June ’08) Attach a \(2\)-disk \(D^2\) to a torus \(S^1\times S^1\) by the attaching map \(e^{2\pi it}\mapsto(e^{2\pi it},e^{2\pi it})\), thinking of the boundary \(\partial D^2\) and each \(S^1\) factor as the unit circle in the complex plane, and let \(X\) be the resulting space. Compute a presentation for the fundamental group of \(X\).
Solution
Let \(a,b\) be the standard generators of \[ \pi_1(T^2)\cong\langle a,b\mid [a,b]=1\rangle. \] The attaching map winds once positively around each circle factor, so its class is \(ab\). Attaching the disk kills this element. Hence \[ \pi_1(X)\cong\langle a,b\mid [a,b],ab\rangle. \] From \(ab=1\) we obtain \(b=a^{-1}\), after which the commutator relation is automatic. Therefore \[ \boxed{\pi_1(X)\cong\mathbb Z}. \]