Homology of a torus with two parallel circles collapsed ($H_0=\mathbb{Z},H_1=\mathbb{Z},H_2={\mathbb{Z}}^2$)

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 $S^1\vee S^2\vee S^2$.