On the Hopf conjecture with symmetry

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20 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)563-593
Number of pages31
JournalGeometry and Topology
Issue number1
StatePublished - Apr 8 2013
Externally publishedYes


  • Grove program
  • Hopf conjecture
  • Positive sectional curvature
  • Steenrod algebra

ASJC Scopus subject areas

  • Geometry and Topology


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