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

2022/11/23 by Nienhaus, Jan · 2 citations
#53C20 #55R40 #55S05 #57S15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.13151

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 T⁴-action on the manifold, improving from previous theorems of Kennard, Wiemeler and Wilking assuming a T⁵-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 T⁶-action.

Cited by

• Decomposition of the Curvature Operator and Applications to the Hopf Conjecture
• On Hopf's conjecture and positive second intermediate Ricci curvature

Related

• On Hopf's conjecture and positive second intermediate Ricci curvature
• Cohomogeneity one manifolds with singly generated rational cohomology
• Topological properties of positively curved manifolds with symmetry
• Splitting of torus representations and applications in the Grove\n symmetry program
• Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16