= Pinched torus
{title2=$T^2/S^1\simeq S^1\vee S^2$}
Collapsing an essential meridian circle of a <torus> to one point gives the <pinched torus>. In the <torus>'s CW structure, the two-cell attaches to two circles by the <commutator>. Collapsing one circle leaves one one-cell and a two-cell attached by a <null-homotopic> word. Thus the quotient is homotopy equivalent to $S^1\vee S^2$. Its cellular chain groups are $\mathbb Z$ in degrees zero, one and two, with zero differentials, so its integral <homology groups> are $H_0=H_1=H_2=\mathbb Z$ and vanish in all higher degrees. This illustrates that collapsing a circle can preserve a top-dimensional homology class even while reducing the first homology rank.
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