The point $x$ on the quotient equator has two preimages. Small discs around them become four half-discs glued along their common diameter, so a neighborhood of $x$ is the cone on a graph with two vertices joined by four edges. The <Excision theorem> and the <local homology from a link> identify
$$
H_i(X,X\setminus\{x\})\cong\widetilde H_{i-1}(L),
$$
where $L$ is this four-edge graph. It is connected and has first Betti number $4-2+1=3$. Hence
$$
H_i(X,X\setminus\{x\})\cong
\begin{cases}
\mathbb Z^3,&i=2,\\
0,&i\ne2.
\end{cases}
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 1 / c / Solution

If a neighborhood of $x$ is a cone on a compact link $L$, excision and the long exact sequence of the cone pair give
$$
H_i(X,X\setminus\{x\})\cong\widetilde H_{i-1}(L).
$$


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 Incoming links: Local homology from a link