On the classification of certain 1-connected 7-manifolds and related problems

TL;DR

This paper classifies certain 7-dimensional manifolds with specific cohomology properties and shows they admit metrics with positive Ricci curvature, advancing understanding of their geometric and topological structure.

Contribution

It provides a classification of 1-connected 7-manifolds with cohomology ring isomorphic to that of  P^2  S^3 and proves their positive Ricci curvature existence.

Findings

Classified 1-connected 7-manifolds with specified cohomology.

Proved existence of positive Ricci curvature metrics on these manifolds.

Extended results to manifolds with cohomology ring of S^2  S^5.

Abstract

We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$ or $H^*(S^2\times S^5)$, this $7$-manifold admits a Riemannian metric with positive Ricci curvature.