(Michigan Sept ’10) Consider the 2-dimensional torus \(T\) and the topological space \[X=T\times[-1,1]/\sim\] where \((x,t)\sim(x',t')\) if either \((x,t)=(x',t')\) or \(t=t'\in\{-1,1\}\). Compute \(H_*(X,\mathbb Z)\).
Solution
The quotient collapses \(T\times\{-1\}\) to one point and \(T\times\{1\}\) to another, so it is precisely the unreduced suspension \(\Sigma T\). Suspension shifts reduced homology by one: \[ \widetilde H_n(\Sigma T;\mathbb Z)\cong \widetilde H_{n-1}(T;\mathbb Z). \] For the torus, \[ H_0(T)=\mathbb Z,\qquad H_1(T)=\mathbb Z^2,\qquad H_2(T)=\mathbb Z, \] and all other homology groups vanish. Since \(\Sigma T\) is connected, we obtain \[ H_n(X;\mathbb Z)\cong \begin{cases} \mathbb Z,&n=0,\\ 0,&n=1,\\ \mathbb Z^2,&n=2,\\ \mathbb Z,&n=3,\\ 0,&n\ge4. \end{cases} \]