  We use the $\eta$ invariants of spin$^c$ Dirac operators to distinguish
connected components of moduli spaces of Riemannian metrics with positive Ricci
curvature. We then find infinitely many non-diffeomorphic five dimensional
manifolds for which these moduli spaces each have infinitely many components.
The manifolds are total spaces of principal $S^1$ bundles over
$\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$ and the metrics are lifted from
Ricci positive metrics on the bases. Along the way we classify 5-manifolds with
fundamental group $\mathbb{Z}_2$ admitting free $S^1$ actions with simply
connected quotients.
