= 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 gives
$$
H_2(Y,Y\setminus\{z\};\mathbb Z)
\cong\widetilde H_1(\Theta_n;\mathbb Z)
\cong\mathbb Z^{n-1}.
$$
When $n>2$, 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.
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