On the Hopf Conjecture with Symmetry

TL;DR

This paper proves the Hopf conjecture for even-dimensional positively curved manifolds with a large symmetry group, using Steenrod algebra actions on cohomology to establish the result.

Contribution

It introduces a novel approach leveraging Steenrod algebra actions to prove the Hopf conjecture under symmetry assumptions.

Findings

Confirmed the Hopf conjecture for manifolds with high-dimensional torus symmetry.

Established a new method using Steenrod algebra on cohomology.

Extended the class of manifolds for which the conjecture holds.

Abstract

The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new tool is the action of the Steenrod algebra on cohomology.