Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 20F / c / Solution

Attaching the cone collapses the circle $A=S^1\times \{x\}$ in the torus. The quotient has the homotopy typeEquivalently, the exact sequence in part (b) kills one of the two generators of $H_1(T^2)$ and preserves the fundamental two-cycle. Therefore