  We show that the moduli space of metrics of nonnegative sectional curvature
on every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. We
also show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy
${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type for
which the moduli space of metrics of positive Ricci curvature has infinitely
many path components. Examples of closed manifolds with finite fundamental
group with these properties were known before only in dimensions $4k+3\geq 7$.
