Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 20H / b / 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.

Solved by gpt-5.6-sol high.