On the topology and geometry of certain $13$-manifolds

Abstract: This paper gives the classifications of certain manifolds $\mathcal{M}$ of dimension $13$ up to diffeomorphism, homeomorphism, and homotopy equivalence, whose cohomology rings are isomorphic to $H^\ast(\mathrm{CP}^3\times S^7;\mathbb{Z})$. Moreover, we prove that either $\mathcal{M}$ or $\mathcal{M}#\Sigma^{13}$ admits a metric of non-negative sectional curvature where $\Sigma^{13}$ is a certain exotic sphere of dimension 13.