I will present an example involving only (non-compact) manifolds without boundary. As far as I know, the analogous problem for closed manifolds is wide open. Nevertheless, the article *Configuration spaces are not homotopy invariant* by Paolo Salvatore and Riccardo Longoni is very much worth looking at (it was published in Topology, volume 44, number 2, 2005, pages 375-380).$\newcommand{\RR}{\mathbb{R}}$$\newcommand{\Conf}{\mathrm{Conf}}$$\newcommand{\To}{\longrightarrow}$

**Proposition:** Assume $n \geq 3$. Let $W$ be a contractible topological $n$-manifold without boundary, and $O \subset W$ an open subspace homeomorphic to $\RR^n$. Then the inclusion of $O$ into $W$ induces homotopy equivalences on all ordered and unordered configuration spaces.

A few consequences of this are:

An embedding $V \to W$ between contractible $n$-manifolds without boundary induces a homotopy equivalence on all ordered and unordered configuration spaces.

If $V$ and $W$ are contractible $n$-manifolds without boundary, then there are homotopy equivalences $\Conf(V,k) \simeq \Conf(W,k)$ between their ordered configuration spaces, and $C_k V \simeq C_k W$ between their unordered configuration spaces.

In particular, the configuration spaces, both ordered and unordered, of the Whitehead manifold are homotopy equivalent to the corresponding configuration spaces of $\RR^3$.

The remainder of this post discusses the proof of the proposition above. Here is a lemma which will be used.

**Lemma:** Let $S$ be any finite subset of $O$. Then the inclusion $O\setminus S \to W\setminus S$ is a weak equivalence.

*Proof*:
Apply the van Kampen theorem to the open cover of $W$ by $O$ and $W\setminus S$. Since $O\setminus S$ is simply connected (since it is equivalent to a wedge sum of $(n-1)$-dimensional spheres), it follows that $\Pi_1(W\setminus S) \simeq \Pi_1(W)$ is trivial.

Now apply the Mayer–Vietoris sequence for homology to the same open cover of $W$ by $O$ and $W\setminus S$. The contractibility of $W$ implies that the inclusion $O\setminus S \to W\setminus S$ is a homology equivalence. Since both spaces are simply connected, this inclusion is also a weak equivalence. ■

Now we prove that the map on ordered configuration spaces $\Conf(O,k) \to \Conf(W,k)$ is a weak equivalence for any $k>0$. Note that we have a map of (horizontal) fibration sequences
$$
\begin{matrix}
O\setminus S & \To & \Conf(O,k) & \To & \Conf(O,k-1) \\
\big\downarrow & & \big\downarrow & & \big\downarrow \\
W\setminus S & \To & \Conf(W,k) & \To & \Conf(W,k-1)
\end{matrix}
$$
where $S$ is any subset of $O$ of size $k-1$. The previous lemma states that the map on the fibres is a weak equivalence. Thus, by induction on $k$, we conclude that the map $\Conf(O,k) \to \Conf(W,k)$ is a weak equivalence.

Now the result for unordered configuration spaces follows since we have a commutative diagram of fibration sequences
$$
\begin{matrix}
\Sigma_k & \To & \Conf(O,k) & \To & C_k O \\
\big\downarrow & & \big\downarrow & & \big\downarrow \\
\Sigma_k & \To & \Conf(W,k) & \To & C_k W
\end{matrix}
$$
which is an equivalence on the fibres and total spaces. We conclude that the map on the base spaces is also a weak equivalence.

for all sufficienty large $k$. $\endgroup$