On Hopf's conjecture and positive second intermediate Ricci curvature

TL;DR

This paper proves that certain high-symmetry, positively curved manifolds of dimension divisible by four have positive Euler characteristic, extending previous results and introducing new topological tools.

Contribution

It extends the Four Periodicity Theorem to broader cases and establishes conditions under which positive second intermediate Ricci curvature implies positive Euler characteristic.

Findings

Manifolds with positive second intermediate Ricci curvature and high torus symmetry have positive Euler characteristic.

Extension of the Four Periodicity Theorem to non-zero degree cohomology cases.

New topological methods for analyzing positively curved manifolds.

Abstract

Hopf conjectured that even-dimensional closed Riemannian manifolds with positive sectional curvature have positive Euler characteristic. The conclusion of the conjecture is known to fail if the positive sectional curvature assumption is relaxed in any number of ways, including to positive second intermediate Ricci curvature. Here we prove that if a manifold with positive second intermediate Ricci curvature has dimension divisible by four and torus symmetry of rank at least ten, then it has positive Euler characteristic. A crucial new tool is a non-trivial extension of the first author's Four Periodicity Theorem to situations where the periodicity of the cohomology does not extend all the way down to degree zero.