Classification for smooth manifolds looking like \mathbbCP³× S⁷

2025/10/01 by Shen, Wen
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.00613

Abstract

In this paper, we classify simply connected closed smooth 13-dimensional manifolds whose cohomology ring is isomorphic to that of \mbCP³× S⁷, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a manifold satisfies certain conditions, either itself or its connected sum with an exotic 13-sphere Σ¹³ admits a Riemannian metric of non-negative sectional curvature. As an additional application of our classification, we classify the diagonal S¹-actions on S⁷× S⁷.

Citations

• Positive Ric₂ curvature on products of spheres and their quotients via intermediate fatness
• PL cobordism and classification of PL manifolds
• On the topology and geometry of certain 13-manifolds

Related

• On the topology and geometry of certain 13-manifolds
• Cohomology rings of certain seven dimensional manifolds
• Enumerating Smooth Structures on ℂP³×\mathbbS^k
• Free circle actions on certain simply connected 7-manifolds
• Fake 13-projective spaces with cohomogeneity one actions