Positive curvature and rational ellipticity

Abstract

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.

References in corpus (4)

• Examples of Riemannian Manifolds with non-negative sectional curvature
• Nonnegatively and Positively curved manifolds
• Positive curvature and rational ellipticity
• An exotic sphere with positive curvature

Cited by in corpus (7)

• Positive curvature and rational ellipticity
• Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16
• Infinite families of manifolds of positive $k^{\mathrm{th}}$-intermediate Ricci curvature with $k$ small
• Positive intermediate Ricci curvature on products of homogeneous spaces
• Soft restrictions on positively curved Riemannian submersions
• An intrinsic curvature condition for submersions over Riemannian manifolds
• Homology versus homotopy in fibrations and in limits