An improved four-periodicity theorem and a conjecture of Hopf with symmetry

TL;DR

This paper proves a version of Hopf's conjecture for positively curved, even-dimensional manifolds with a $T^4$-symmetry, using an improved four-periodicity theorem and rational cohomology classification.

Contribution

It introduces an enhanced four-periodicity theorem and classifies fixed point components, extending previous results to manifolds with lower symmetry assumptions.

Findings

Proves Hopf's conjecture under $T^4$-symmetry.

Classifies fixed point components via rational cohomology.

Provides a classification for manifolds with $T^6$-symmetry.

Abstract

In the 1930s, H. Hopf conjectured that a closed, even-dimensional manifold of positive sectional curvature has positive Euler characteristic. We show this under the additional assumption of an isometric $T^4$-action on the manifold, improving from previous theorems of Kennard, Wiemeler and Wilking assuming a $T^5$-action. More specifically, this is achieved by giving a rational cohomology classification of possible fixed point components. The main new tool is an improvement on the four-periodicity theorem originally developed by Kennard through the use of characteristic class theory. As a second application we give a rational cohomology classification of closed positively curved even-dimensional manifolds without odd rational cohomology that admit an isometric $T^6$-action.