# Stable range of some classifying spaces and iterated loop spaces

Galatius (in his talk) has made very interesting remarks about the stable range of some classifying spaces of groups. To be more concrete, I will mention two examples to illustrate his (?) point of view.

Example 1

Let $Br_{n}$ the $n$-braided groups then the integral homology $H_{\ast}(BBr_{n},\mathbf{Z})$ is isomorphic (in stable range) to $H_{\ast}(\Omega^{2}_{0}S^{2},\mathbf{Z})$ where the $\Omega^{2}_{0}S^{2}$ is the connected component of double loop space of the sphere. $$BBr_{\infty}\simeq_{H_{\ast}} \Omega^{2}_{0}S^{2}$$

Example 2

In the case of the group of automorphisms of the free group with $n$-generators $Aut(F_{n})$ he proved that there is a homology isomorphism (in stable range) between $H_{\ast}(BAut(F_{n}),\mathbf{Z})$ and $H_{\ast}(\Omega^{\infty}_{0}S^{\infty},\mathbf{Z})$ i.e., $$B[colim_{n} Aut(F_{n})]\simeq_{H_{\ast}}\Omega^{\infty}_{0}S^{\infty}$$

Question: Is there examples of a sequence of (topological) groups $\dots\rightarrow G_{n}\rightarrow G_{n+1}\dots$ such that we have a homology isomorphism between $$BG_{\infty}\simeq_{H_{\ast}}\Omega^{i}_{0}X$$ where $X$ is not a loop space and $2<i<\infty$. In particular is there a sequence of (topological) groups $\dots\rightarrow G_{n}\rightarrow G_{n+1}\dots$ such $$BG_{\infty}\simeq_{H_{\ast}}\Omega^{n}_{0}S^{n}$$

Examples with a geometric meaning will be appreciated. If I wrote something wrong, do please correct me.