Volume 17, issue 1 (2013)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On the Hopf conjecture with symmetry

Lee Kennard

Geometry & Topology 17 (2013) 563–593
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