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Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this is only a piecewise projective linear map. I don't know why (maybe, I didn't get the right statement). Any references or explanations will be welcomed.

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