Algebraic & Geometric Topology

Positive curvature and rational ellipticity

Manuel Amann and Lee Kennard

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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.

Article information

Algebr. Geom. Topol., Volume 15, Number 4 (2015), 2269-2301.

Received: 9 June 2014
Revised: 22 October 2014
Accepted: 24 November 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 57N65: Algebraic topology of manifolds 55P62: Rational homotopy theory

positive curvature rational ellipticity torus symmetry Euler characteristic Wilhelm conjecture Halperin conjecture Hopf conjecture


Amann, Manuel; Kennard, Lee. Positive curvature and rational ellipticity. Algebr. Geom. Topol. 15 (2015), no. 4, 2269--2301. doi:10.2140/agt.2015.15.2269.

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