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 $\hat{I}{\text{c◯}}_8^{Spin}\left(K_2\right)$ and $Ω_{8}^{Spin}\left(K_{2};\mathrm{pr}_{1}^{*}γ^{1}\right)$ that are required in the deduction of $s$-invariants.

43 pages. Comments are welcome!