Nonconnected moduli spaces of positive sectional curvature metrics

For a closed manifold $M$ let $\Re _{{\text{sec}}}^ + (M)$ (resp. $\Re _{{\text{Ric}}}^ + (M)$) be the space of Riemannian metrics on $M$ with positive sectional (resp. Ricci) curvature and let ${\text{Diff}}(M)$ be the diffeomorphism group of $M$, which acts on these spaces. We construct examples of $7$-dimensional manifolds for which the moduli space $\Re _{{\text{sec}}}^ + (M)/{\text{Diff}}(M)$ is not connected and others for which $\Re _{{\text{Ric}}}^ + (M)/{\text{Diff}}(M)$ has infinitely many connected components. The examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by Aloff and Wallach, on which $SU(3)$ acts transitively, respectively a family of positive Einstein metrics constructed by Wang and Ziller on homogeneous spaces, on which $SU(3) \times SU(2) \times U(1)$ acts transitively.