Geometry & Topology

On the Hopf conjecture with symmetry

Lee Kennard

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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.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 563-593.

Received: 29 May 2012
Revised: 17 November 2012
Accepted: 20 December 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 55S10: Steenrod algebra

positive sectional curvature Hopf conjecture Grove program Steenrod algebra


Kennard, Lee. On the Hopf conjecture with symmetry. Geom. Topol. 17 (2013), no. 1, 563--593. doi:10.2140/gt.2013.17.563.

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