Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 114 / 1 / c / Solution

At a point of the open two-cell, the local link is a circle, so local homology in degree two has rank one. At a point of $Z$, a neighborhood consists of $n$ half-discs sharing their boundary interval; its link is the Theta graph ${\Theta }_n$. The local homology from a link calculation givesWhen $n>2$, this local rank characterizes the points of $Z$. Since local homology is invariant under a homeomorphism, every homeomorphism satisfies $f(Z)=Z$, and in particular $f(Z)\subset Z$.

For $n=2$, the two pages form an ordinary disc neighborhood, and $Y$ is ${\mathbb{RP}}^2$. The distinguished circle $Z$ is a projective line, but a projective linear homeomorphism can carry it to a different projective line. Thus the conclusion does not hold.