On the homology theory of the closed geodesic problem

Let \(\Lambda X\) be the free loop space on a simply connected finite \(CW\)-complex \(X\) and \(\beta_{i}(\Lambda X;\Bbbk)\) be the cardinality of a minimal generating set of \(H^{i}(\Lambda X;\Bbbk)\) for \(\Bbbk\) to be a commutative ring with unit. The sequence \( \beta_{i}(\Lambda X;\Bbbk) \) g...

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Bibliographic Details
Published in:arXiv.org 2011-11
Main Author: Samson Saneblidze
Format: Article
Language:English
Subjects:
Geodesy
Homology
Online Access:Get full text
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Summary:Let \(\Lambda X\) be the free loop space on a simply connected finite \(CW\)-complex \(X\) and \(\beta_{i}(\Lambda X;\Bbbk)\) be the cardinality of a minimal generating set of \(H^{i}(\Lambda X;\Bbbk)\) for \(\Bbbk\) to be a commutative ring with unit. The sequence \( \beta_{i}(\Lambda X;\Bbbk) \) grows unbounded if and only if \(\tilde {H}^{\ast}(X;\Bbbk)\) requires at least two algebra generators. This in particular answers to a long standing problem whether a simply connected closed smooth manifold has infinitely many geometrically distinct closed geodesics in any Riemannian metric.
ISSN:2331-8422