Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
Does $PM$ satisfy fixed point property?
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$\begingroup$ Sometimes yes (e.g., $k=0$), sometimes no (e.g., $k=n$). $\endgroup$– Alex DegtyarevCommented Mar 4, 2015 at 9:26
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$\begingroup$ @AlexDegtyarev For non critical $k$? $\endgroup$– Ali TaghaviCommented Mar 4, 2015 at 9:29
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$\begingroup$ What is "fixed point property"? $\endgroup$– SashaCommented Mar 4, 2015 at 11:28
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$\begingroup$ @Sasha $X$ has fixed point property if for every continuous map $f$ on $X$ there is a $b\in X$ with $f(b)=0$. $\endgroup$– Ali TaghaviCommented Mar 4, 2015 at 11:39
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1$\begingroup$ I think you meant $f(b)=b$ here. $\endgroup$– Alex DegtyarevCommented Mar 4, 2015 at 11:50
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1 Answer
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Usually no (check the precise dimensions yourself). In fact, just the map $M\mapsto GM$ for $G\in GL(n)$ typically has no fixed points. For $GM=\lambda M$ implies that the columns of $M$ are eigenvectors of $G$ (with the same eigenvalue $\lambda$). Thus, it suffices to assume that $G$ has no eigenspace of dimension $\ge k$.
answered Mar 4, 2015 at 11:48