Find a cell structure for the quotient space obtained by identifying two distinct points \(a,b\) in a 2-torus to a third point \(c\) in a 2-sphere, and compute a presentation for the fundamental group of this space.
Solution
Let \(Q\) denote the quotient space.
1. The torus has a CW structure with \(0\)-cells \(a,b\), \(1\)-cells \(r\colon a\to b\), \(s\colon b\to a\), \(t\colon a\to a\), and one \(2\)-cell attached along \((rs)t(rs)^{-1}t^{-1}\).
Take the standard CW structure on \(T^2\) with vertex \(a\), loops \(x,t\), and one \(2\)-cell attached along \(xtx^{-1}t^{-1}\), chosen so that \(b\) lies in the interior of the \(x\)-circle. Subdividing \(x\) at \(b\) into edges \(r\) and \(s\) writes \(x\) as the edge path \(rs\), which gives the stated attaching word.
2. \(Q\) is a CW complex with one \(0\)-cell \(v\), loops \(r,s,t\) at \(v\), a \(2\)-cell \(e_T\) attached along \((rs)t(rs)^{-1}t^{-1}\), and a \(2\)-cell \(e_S\) attached by the constant map at \(v\).
\(S^2=D^2/\partial D^2\) has a CW structure with \(0\)-cell \(c\) and one \(2\)-cell attached by the constant map. The quotient identifies only the three \(0\)-cells \(a,b,c\) to \(v\), so \(r,s,t\) become loops at \(v\), and each \(2\)-cell keeps its attaching map from step 1 or from the sphere.
3. \(\pi_1(Q,v)\cong\langle r,s,t\mid(rs)t(rs)^{-1}t^{-1}\rangle\).
The \(1\)-skeleton is a wedge of three circles, with fundamental group free on \(r,s,t\). By the Seifert–van Kampen theorem, attaching \(e_T\) imposes its attaching word as a relation, and the constant attaching loop of \(e_S\) imposes the trivial relation.
4. Q.E.D.
Step 2 gives the cell structure. In the presentation of step 3, substitute \(x=rs\), \(y=t\), \(z=r\); then \(s=z^{-1}x\), so \(x,y,z\) generate freely subject to the single relation \([x,y]=1\). Hence \[\pi_1(Q)\cong\langle x,y,z\mid[x,y]\rangle\cong\langle x,y\mid[x,y]\rangle*\langle z\rangle\cong\ZZ^2*\ZZ.\]