Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups
type: Item
instance of: scholarly article
title: Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups
zbMATH Open document ID: 1547.55013
author: Ruizhi Huang
author: Stephen Theriault
publication date: 2024-07-05T00:00:00Z
review text: It is a classical problem to study the growth of the homology groups of the free or the based loop space of a closed manifold. A question that is studied often in the literature is to understand the growth of the rational homology groups of a free loop space using methods from rational homotopy theory. In particular, it is a conjecture by Vigué-Porrier that the rational homology groups of a rationally hyperbolic space grow exponentially.\N\NInstead of using methods of rational homotopy theory, the authors of the present article use integral methods to obtain results about the exponential growth of the rational homology of the free loop spaces of certain classes of spaces. Because of this integral approach, the authors can then also state a result about the growth of the \(p\)-torsion in the homotopy groups of a particular class of spaces. Some of the results that are obtained in this article were already proven by \textit{Y. Félix} et al. [Pure Appl. Math. Q. 9, No. 1, 167--187 (2013; Zbl 1300.55016); Ann. Inst. Fourier 67, No. 6, 2519--2531 (2017; Zbl 1409.55009)].\N\NIn order to state the main results of this article, we need to recall the notions of \textit{exponential growth} of a graded vector space. Note that these definitions go back to [Zbl 1300.55016]. Let \(V = \{ V_i\}_{i\geq 0}\) be a graded vector space. Define the \textit{log index} of \(V\) to be \N\[\N\mathrm{log\,index}(V) = \limsup_{i\to\infty}\frac{\log(\mathrm{dim}(V_i))}{i}.\N\]\NThe graded vector space \(V\) is said to have \textit{good exponential growth} if \(0<\mathrm{log\,index}(V) < \infty\) and if for each \(\lambda > 1\) there exists a sequence \((n_k)_{k\geq 1}\) such that \(n_i < n_{i+1} < \lambda n_i\) and such that \(\mathrm{dim}(V_i) = e^{\alpha_i n_i}\) with \(\alpha_i \to \mathrm{log\,index}(V)\).\N\NOne can now define the notion of \textit{good exponential growth} for the rational homology of a free loop space. Let \(X\) be a simply connected space with rational homology of finit…
reviewed by: Maximilian Stegemeyer
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 7877215
zbMATH Keywords: exponential growth
zbMATH Keywords: free loop space
zbMATH Keywords: homotopy exponent
zbMATH Keywords: Moore's conjecture
MaRDI profile type: Publication
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published in: Annales de l’institut Fourier
DOI: 10.5802/AIF.3627
description: scientific article; zbMATH DE number 7877215