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

TL;DR

This paper classifies certain 13-dimensional manifolds with specific cohomology rings and explores their geometric properties, including the existence of non-negative curvature metrics, revealing new insights into their topology and geometry.

Contribution

It provides a classification of 13-manifolds with a given cohomology ring and establishes the existence of non-negative curvature metrics on these manifolds or their connected sums with exotic spheres.

Findings

Classified 13-manifolds up to diffeomorphism, homeomorphism, and homotopy equivalence.

Proved existence of non-negative sectional curvature metrics on these manifolds or their connected sums.

Identified conditions involving exotic spheres in the geometric structure of these 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.