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

Abstract

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 $\left(\mathbb{C}P^{1}\times\mathbb{C}P^{2}\right)\#\mathbb{C}P^{3}$ 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 $\Omega_{8}^{Spin}\left(K_{2}\right)$ and $\Omega_{8}^{Spin}\left(K_{2};\mathrm{pr}_{1}^{*}\gamma^{1}\right)$ that are required in the deduction of $s$-invariants.