Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 15, Number 4 (2015), 2269-2301.
Positive curvature and rational ellipticity
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.
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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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