Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups (Q6567983)

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 finite type and assume that \(\mathrm{log\,index}(\mathrm{H}_{\bullet}(\Omega X;\mathbb{Q}))\in(0,\infty)\), where \(\Omega X\) is the based loop space of \(X\). If \(\mathcal{L}X\) is the free loop space of \(X\), then it holds that \( \mathrm{log\,index}(\mathrm{H}_{\bullet}(\mathcal{L} X;\mathbb{Q}) \leq \mathrm{log\,index}(\mathrm{H}_{\bullet}(\Omega X;\mathbb{Q})\). We say that the free loop space \(\mathcal{L}X\) has \textit{good exponential growth} if \(\mathrm{log\,index}(\mathrm{H}_{\bullet}(\mathcal{L} X;\mathbb{Q}) = \mathrm{log\,index}(\mathrm{H}_{\bullet}(\Omega X;\mathbb{Q}) \) and if \(\mathrm{H}_{\bullet}(\Omega X;\mathbb{Q})\) has controlled exponential growth.\N\NThe main result of the article is the following.\N\N\textbf{Theorem} Let \(\Sigma A \xrightarrow[]{f} Y \xrightarrow[]{h} Z\) be a homotopy cofibration with \(A,Y\) and \(Z\) being simply connected finite CW-complexes. Assume that \(A\) and \(Z\) are not rationally contractible and that \(\Omega h\) has a homotopy inverse. Then the following statements hold\N\begin{itemize}\N\item[1.] if \(\mathrm{log\,index}(\pi_{\bullet}(Z)) < \mathrm{log\,index}(\pi_{\bullet}(Y)) \) and \(\mathrm{log\,index}(\mathrm{H}_{\bullet}(\Omega Y;\mathbb{Q}))\in (0,\infty)\), then \(\mathcal{L}Y\) has good exponential growth.\N\item[2.] if \(Z\) is rationally hyperbolic with a finitely generated rational homotopy Lie algebra, then \(\mathcal{L}Y\) has good exponential growth.\N\item[3.] if \(\mathrm{H}_{\bullet}(Y;\mathbb{Z})\) has no \(p\)-torsion for a sufficiently large prime \(p\), then \(Y\) is rationally hyperbolic and \(\mathbb{Z}/p^r\)-hyperbolic for all \(r\geq 1\).\N\end{itemize}\NNote that for the last part of the theorem, a \(p\)-local space \(X\) is called \(\mathbb{Z}/p^r\)-\textit{hyperbolic} if the number of \(\mathbb{Z}/p^r\)-summands in \(\pi_*(X)\) grows exponentially, see Definition 1.4 of the article for the precise definition. Because of the last part of the main theorem, the authors conjecture that there is a strong connection between rational hyperbolicity and \(\mathbb{Z}/p^r\)-hyperbolicity. The authors state the conjecture that rational hyperbolicity of a finite simply connected CW-complex implies \(\mathbb{Z}/p^r\)-hyperbolicity for all primes \(p\) and all \(r\geq 1\), see Conjecture 1.6.