The homology of a free loop space
type: Item
instance of: scholarly article
title: The homology of a free loop space
zbMATH Open document ID: 0666.55011
author: Stephen Halperin
author: Micheline Vigué-Poirrier
published in: Pacific Journal of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: Denote by \(X^{S^ 1}\) the space of all continuous maps from the circle into a simply connected finite CW complex X. Theorem: Let \({\mathbb{K}}\) be a field and suppose that either char \({\mathbb{K}}>\dim X\) or that X is \({\mathbb{K}}\)-formal. Then the Betti numbers \(b_ q=\dim H_ q(X^{S^ 1};{\mathbb{K}})\) are uniformly bounded above if and only if the \({\mathbb{K}}\)- algebra H*(X;\({\mathbb{K}})\) is generated by a single cohomology class. Corollary: If, in addition, X is a smooth closed manifold and \({\mathbb{K}}\) is as in the theorem, and if H*(X;\({\mathbb{K}})\) is not generated by a single class then X has infinitely many distinct closed geodesics in any Riemannian metric.
Mathematics Subject Classification ID: 55R20
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4090367
zbMATH Keywords: free loop space
zbMATH Keywords: Hochschild homology
zbMATH Keywords: infinitely many distinct closed geodesics
MaRDI profile type: Publication
full work available at URL: pjm.1991.147.311
OpenAlex ID: W1979169306
DOI: 10.2140/PJM.1991.147.311
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description: scientific article; zbMATH DE number 4090367