I am looking for a proof of the following result for $n \geq 1$, $n \neq 4$.
Let $(K,h)$ be a PL structure on $\mathbb{R}^n$ (i.e. a locally finite simplicial complex $K$ and a homeomorphism $h : |K| \rightarrow \mathbb{R}^n$ such that the star of any simplex of $K$ is homeomorphic to a sphere).
Let $0 < \epsilon < a < b$ be three numbers and let $O_n:=\{x \in \mathbb{R}^n : ||x|| < 1\}$.
If $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a homeomorphism, there exists a homeomorphism $g : \mathbb{R}^n \rightarrow \mathbb{R}^n$ which is PL relative to $(K,h)$ (i.e. there exists a subdivision $K_1$ of $K$ such that for each closed simplex $\sigma \in K_1$, $h^{-1} \circ g \circ h$ restricted to $\sigma$ is the restriction of an affine map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ on $\sigma$) and is such that:
$g|_{f[(a-\epsilon)O_n]} = Id$
$g|_{f[(b+\epsilon)O^c_n]} = Id$
$g[f(aO_n)] \supseteq f(bO_n)$
the angle between $g^{-1}(y)$ and $f^{-1}(y)$ is $< \epsilon$ for all $y \in \mathbb{R}^n$
Any idea on a paper that would prove this? I think this is somehow related to the generalized Poincare conjecture in the case of PL manifolds but this is more precise than the statement of this conjecture.
I just need to quote a reference that would handle this result. Thank you in advance!