Analogue of $O(n) \subset GL(n,\mathbb{R})$ in $Diff, PL, TOP$ Here is a way to prove that the homotopy groups $\pi_i(GL(n,\mathbb{R}))$ are all finitely generated: $GL(n,\mathbb{R})$ deformation retracts to $O(n)$, which is a compact manifold, hence has finitely generated homology groups. Now $O(n)$ is also a Lie group and so is simple, hence by Serre's mod $\mathcal{C}$ theorem, $\pi_i(GL(n,\mathbb{R}))=\pi_i(O(n))$ are finitely generated.
I am interested to know whether an analogue of this situation exists in other categories. More precisely, let $X$ be:


*

*the group of diffeomorphisms of $\mathbb{R}^n$, or 

*the group of PL isomorphisms of $\mathbb{R}^n$, or

*the group of homeomorphisms of $\mathbb{R}^n$


(all endowed with compact-open topology as usual)
Does there exists $Y \subset X$ such that $X$ deformation retracts to $Y$, and $Y$ is a compact CW complex, preferably also a topological group?
The reason I am asking this is that I want to understand why $\pi_i(X)$ are finitely generated for the above $X$'s, yet these seem to be often stated as corollaries of deep results that calculate the groups explicitly. If I simply need to know finitely generated-ness, is there a more elementary way? 

Theorem A from this paper of Antonelli, Burghelea and Kahn, The non-finite homotopy of some diffeomorphism groups gives a negative answer to your question in the smooth setting.
