String homology, and closed geodesics on manifolds which are elliptic spaces
type: Item
instance of: scholarly article
title: String homology, and closed geodesics on manifolds which are elliptic spaces
zbMATH Open document ID: 1382.55007
author: John McCleary
author: John D. S. Jones
published in: Algebraic \& Geometric Topology
publication date: 2016-11-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1409.8643
Mathematics Subject Classification ID: 55P50
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55T05
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 6653754
zbMATH Keywords: string homology
zbMATH Keywords: closed geodesics
zbMATH Keywords: elliptic spaces
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
OpenAlex ID: W1896478298
arXiv ID: 1409.8643
DOI: 10.2140/AGT.2016.16.2677
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review text: Let \(M\) be a simply connected closed smooth manifold of dimension \(n\geq 2\). It is conjectured that if, for some field \({\Bbbk}\), the cohomology algebra \(H^*(M;{\Bbbk})\) is generated by at least two generators then for any Riemannian metric on \(M\) there exist infinitely many geometrically distinct closed geodesics. This conjecture, also called the closed geodesic problem, has been solved by many authors under additional hypotheses. Here the authors prove this conjecture for the prime fields \(\mathbb{F}_p\), \(p\geq 2\) and manifolds \(M\) which are \(\mathbb{F}_p\)-elliptic spaces. NEWLINENEWLINEEllipticity is a very severe restriction on topological spaces while many of the spaces which play a role in geometry are elliptic. For example, simply connected compact homogeneous spaces, simply connected positively curved closed manifolds, fibrations built out of elliptic spaces, et cetera. The proof uses string topology as well as older results on elliptic connected graded Hopf algebras.
description: scientific article; zbMATH DE number 6653754