(Michigan Sept ’08) Compute the homology of the space formed as the union of the unit sphere \[\{(x,y,z)\mid x^2+y^2+z^2=1\}\] and the closed interval along the \(z\)-axis from \((0,0,-1)\) to \((0,0,1)\).
Solution
Let \(I\) be the added diameter joining the south and north poles of \(S^2\). Attaching a \(1\)-cell between two distinct points of a connected CW complex adds one free circle up to homotopy. Concretely, choose an arc \(J\subset S^2\) from one pole to the other; the union \(I\cup J\) is a circle, and collapsing a complementary maximal tree in a compatible CW structure yields \[ S^2\cup I\simeq S^2\vee S^1. \] Consequently \[ H_n(S^2\cup I;\mathbb Z)\cong \begin{cases} \mathbb Z,&n=0,\\ \mathbb Z,&n=1,\\ \mathbb Z,&n=2,\\ 0,&n\ge3. \end{cases} \] The same computation follows from Mayer–Vietoris after thickening \(S^2\) and the interval to open neighborhoods whose intersection has two contractible components.