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Positive curvature and rational ellipticity

Manuel Amann and Lee Kennard

Algebraic & Geometric Topology 15 (2015) 2269–2301

Simply connected manifolds of positive sectional curvature are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, ie to have only finitely many non-zero rational homotopy groups. In this article, we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include an upper bound on the Euler characteristic and new evidence for a couple of well-known conjectures due to Hopf and Halperin. We also prove a conjecture of Wilhelm for even-dimensional manifolds whose rational type is one of the known examples of positive curvature.

positive curvature, rational ellipticity, torus symmetry, Euler characteristic, Wilhelm conjecture, Halperin conjecture, Hopf conjecture
Mathematical Subject Classification 2010
Primary: 53C20
Secondary: 57N65, 55P62
Received: 9 June 2014
Revised: 22 October 2014
Accepted: 24 November 2014
Published: 10 September 2015
Manuel Amann
Fakultät für Mathematik
Institut für Algebra und Geometrie
Karlsruher Institut für Technologie
Englerstraße 2
D-76131 Karlsruhe
Lee Kennard
Department of Mathematics
University of Oklahoma
Norman, OK 73019