(June ’12) Let \(X\) be the space obtained from the torus \(T^2=S^1\times S^1\) by gluing in a disk \(D^2\) along its boundary \(\partial D^2=S^1\) using the map \(\alpha:S^1\to T^2\) given by \(z\mapsto(z,(1,0))\) for \(z\in S^1\).
Find \(\pi_1(X)\).
Give a CW complex for \(X\).
Solution
Use the standard CW structure on the torus: one \(0\)-cell, two \(1\)-cells \(a,b\), and one \(2\)-cell attached by \[ aba^{-1}b^{-1}. \] The new disk is a second \(2\)-cell whose boundary is attached along the \(a\)-circle. Thus a CW structure for \(X\) has one \(0\)-cell, \(1\)-cells \(a,b\), and two \(2\)-cells with attaching words \[ [a,b]\quad\text{and}\quad a. \] Van Kampen gives \[ \pi_1(X)\cong\langle a,b\mid [a,b],a\rangle\cong\langle b\rangle\cong\mathbb Z. \] This answers both parts: the displayed cells give the requested CW complex, and its fundamental group is infinite cyclic.