The local hyperbolicity of \(\mathbf{A}_n^2\)-complexes (Q2214758)

A \(p\)-local complex \(X\) is called \(\mathbb{Z}/p^i\)-hyperbolic (resp. \(p\)-hyperbolic) if the number of \(\mathbb{Z}/p^i\)-summands in \(\pi_*(X)\) (resp. the \(p\)-primary torsion part of \(\pi_*(X)\)) has exponential growth. Let us write \(C^{n+2}_{\eta}=S^n\cup_{\eta}e^{n+2}=\Sigma^{n-2}\mathbb{C}\mathrm{P}^2\), \(C^{n+2,s}=(S^n\vee S^{n+1})\cup_{\binom{\eta}{2^s}}e^{n+2}\), \(C^{n+2}_r=S^n\cup_{(2^r,\eta)}C(S^{n+1}\vee S^{n+1})\), and \(C^{n+2,s}_r= (S^n\vee S^{n+1})\cup_{\tiny \begin{pmatrix} 2^r, & \eta \\ 0, & 2^s \end{pmatrix}} C(S^n\vee S^{n+1})\); these are the elementary Chang complexes introduced in [\textit{S.-C. Chang}, Proc. R. Soc. Lond., Ser. A 202, 253--263 (1950; Zbl 0041.10201)]. In this paper, the authors study the asymptotic behavior of the \(p\)-primary part of the homotopy groups of simply connected finite \(p\)-local complexes. In particular, they consider this problem for the Chang complexes \(C\). They show that the Chang complexes \(C\) are \(\mathbb{Z}/2^i\)-hyperbolic for \(C=C^{n+2,r}_r\) \((n\geq 4)\) with \(i=1,r,r+1\). Moreover, they also show that the Chang complex \(C\) is \(\mathbb{Z}/2\)-hyperbolic if \(\mu_C=1\) and \(\mathbb{Z}/2^i\)-hyperbolic if \(\mu_C>1\) with \(i=1,\mu_C,\mu_C+1,\) where \(n\geq 4\) and \(\mu_C\) denotes the number given by \[ \mu_C= \begin{cases} 1 & \text{ if }C=C^{n+2}_{\eta} \\ r & \text{ if }C=C^{n+2}_r \\ s & \text{ if }C=C^{n+2,s} \\ \min\{r,s\} & \text{ if }C=C^{n+2,s}_r \text{ and }s\not= r \end{cases} \] As an application they prove that if \(A\) is an \(\mathbb{A}^2_n\)-complex \((n\geq 4)\) and it is not a sphere nor a contractible space, then \(A\) is \(\mathbb{Z}/p\)-hyperbolic.