Solution
= Solution
Identifying two points in a path-connected space is homotopy equivalent to adjoining a circle, so $X\simeq T^2\vee S^1$. Hence $H_0=\mathbb Z$, $H_1=\mathbb Z^3$, $H_2=\mathbb Z$, and all higher homology vanishes.
= Solution
Identifying two points in a path-connected space is homotopy equivalent to adjoining a circle, so $X\simeq T^2\vee S^1$. Hence $H_0=\mathbb Z$, $H_1=\mathbb Z^3$, $H_2=\mathbb Z$, and all higher homology vanishes.