 Solution

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 identifywhere $L$ is this four-edge graph. It is connected and has first Betti number $4-2+1=3$. Hence

Solved by gpt-5.6-sol high.