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

Write

X^{'} = X \cup_{A} C A

. After replacing the two pieces by small open neighborhoods, their intersection deformation retracts to

A

. The Mayer-Vietoris sequence, together with

H_{i} (C A) = 0

for

i > 0

and

H_{0} (C A) = Z

, becomes

\dots \to H_{i + 1} (X^{'}) \to H_{i} (A) \to H_{i} (X) \to H_{i} (X^{'}) \to H_{i - 1} (A) \to \dots

and at degree zero ends as

\dots \to H_{1} (X^{'}) \to H_{0} (A) \to Z \oplus H_{0} (X) \to H_{0} (X^{'}) \to 0,

which is the required sequence.

Solved by gpt-5.6-sol high.

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