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arXiv · 2211.13151

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

Jan Nienhaus
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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.

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Jan Nienhaus. 2022-11-23. An improved four-periodicity theorem and a conjecture of Hopf with symmetry. https://arxiv.org/abs/2211.13151

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