On the Hopf conjecture with symmetry

Kennard, Lee

doi:10.2140/gt.2013.17.563

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On the Hopf conjecture with symmetry

Lee Kennard

Geometry & Topology 17 (2013) 563–593

DOI: 10.2140/gt.2013.17.563

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.

Keywords

positive sectional curvature, Hopf conjecture, Grove program, Steenrod algebra

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 55S10

References

Publication

Received: 29 May 2012

Revised: 17 November 2012

Accepted: 20 December 2012

Published: 8 April 2013

Proposed: John Lott

Seconded: Dmitri Burago, Steve Ferry

Authors

Lee Kennard
Department of Mathematics University of California, Santa Barbara Santa Barbara, CA 93106-3080 USA
http://www.math.ucsb.edu/~kennard

Volume 17, issue 1 (2013)