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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 type

X^{'} ≃ S^{1} \lor S^{2} .

Equivalently, the exact sequence in part (b) kills one of the two generators of

H_{1} (T^{2})

and preserves the fundamental two-cycle. Therefore

H_{i} (X^{'}) ≅ {Z, 0, i = 0, 1, 2, i > 2.

Solved by gpt-5.6-sol high.

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