= Homology of a torus with two parallel circles collapsed
{title2=$H_0=\mathbb Z,\quad H_1=\mathbb Z,\quad H_2=\mathbb Z^2$}
Collapse two distinct parallel circles of a <torus> to two separate points. Cutting between the circles gives two annuli, each of which becomes a two-sphere with the two collapsed points as its poles. The resulting <CW complex> has two vertices, two edges connecting them, and two two-cells with zero cellular boundary. Its homology is as displayed, with all higher groups zero; it is homotopy equivalent to $S^1\vee S^2\vee S^2$.
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