= Fixed-point-free complex-structure map on odd-dimensional real projective space
{title2=$[x]\mapsto[Jx]$}
= Fixed-point-free complex-structure maps on odd-dimensional real projective space
{synonym}
Choose a <complex structure> $J$ on $\mathbb R^{2m}$, with $J^2=-I$. Its induced map on $\mathbb RP^{2m-1}$ has no fixed point: $Jx=\lambda x$ for a real nonzero vector would give $\lambda^2=-1$. The maps $\cos t\,I+\sin t\,J$, for $0\leq t\leq\pi/2$, give a <homotopy> from the identity to $J$, because they are invertible throughout. Thus this fixed-point-free map is homotopic to the identity.
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