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Positive curvature and
rational ellipticity
Manuel Amann and Lee Kennard |
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Algebraic & Geometric Topology 15 (2015) 2269–2301
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Abstract |
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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. |
Keywords
positive curvature, rational ellipticity, torus symmetry,
Euler characteristic, Wilhelm conjecture, Halperin
conjecture, Hopf conjecture
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Mathematical Subject Classification 2010
Primary: 53C20
Secondary: 57N65, 55P62
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References |
Publication
Received: 9 June 2014
Revised: 22 October 2014
Accepted: 24 November 2014
Published: 10 September 2015
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Authors |
| Manuel Amann | |
| Fakultät für Mathematik Institut für Algebra und Geometrie Karlsruher Institut für Technologie Englerstraße 2 D-76131 Karlsruhe Germany |
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| http://www.math.kit.edu/iag7/~amann | |
| Lee Kennard | |
| Department of Mathematics University of Oklahoma Norman, OK 73019 USA |
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