On $7$-manifolds with $b_2=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics

We derive the $s$-invariants of certain simply connected $7$-manifolds whose second homology groups are isomorphic to ${\mathbb{Z}}^2$. We apply the $s$-invariants to give a partial classification of simply connected total spaces of circle bundles over $(\mathbb{C}{P}^1\times \mathbb{C}{P}^2)\mathrm{\#}\mathbb{C}{P}^3$ up to diffeomorphism. As an application, we show that there is a simply connected $7$-manifold whose space and moduli space of positive Ricci curvature metrics both have infinitely many path components. We also determine bordism groups ${\mathrm{\Omega }}_8^{Spin}\left(K_2\right)$ and ${\mathrm{\Omega }}_8^{Spin}(K_2;{\mathrm{p}\mathrm{r}}_1^{\ast }{\gamma }^1)$ that are required in the deduction of $s$-invariants.