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Use the usual cell structure on the torus: one vertex, two one-cells $a,b$, and one two-cell attached by the word $ab{a}^{-1}{b}^{-1}$. Take the meridian being collapsed as $a$. The quotient has one vertex, the surviving circle $b$, and a two-cell whose attaching word reduces to $b{b}^{-1}$ and is null-homotopic. Attaching a two-cell to a circle by a null-homotopic loop gives a space homotopy equivalent to $S^1\vee S^2$.

Alternatively its cellular chain groups are $C_2=C_1=C_0=\mathbb{Z}$, and both boundary maps vanish: the two-cell traverses $b$ once in each direction, and the one-cell's endpoints coincide. Thus the integral homology groups areThe pinched torus is connected, so reduced $H_0$ is zero.

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.