Geometry & Topology

On the Hopf conjecture with symmetry

Lee Kennard

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732532

Digital Object Identifier
doi:10.2140/gt.2013.17.563

Mathematical Reviews number (MathSciNet)
MR3039770

Zentralblatt MATH identifier
1267.53038

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

Keywords
positive sectional curvature Hopf conjecture Grove program Steenrod algebra

Citation

Kennard, Lee. On the Hopf conjecture with symmetry. Geom. Topol. 17 (2013), no. 1, 563--593. doi:10.2140/gt.2013.17.563. https://projecteuclid.org/euclid.gt/1513732532


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