## Algebraic & Geometric Topology

### Positive curvature and rational ellipticity

#### Abstract

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

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

Dates
Revised: 22 October 2014
Accepted: 24 November 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841003

Digital Object Identifier
doi:10.2140/agt.2015.15.2269

Mathematical Reviews number (MathSciNet)
MR3402341

Zentralblatt MATH identifier
1325.53044

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841003

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