Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ for definiteness and simplicity. I am not particularly interested in any specific value of $k$.$\newcommand{\Diff}{\operatorname{Diff}}$$\newcommand{\RR}{\mathbb{R}}$

It is fairly well-known that the space $\Diff(M)$ of $C^1$ diffeomorphisms of a closed smooth manifold $M$ is homotopy equivalent to a CW-complex. Here are the relevant facts:

• $\Diff(M)$ is a Banach manifold modelled locally on the space of $C^1$ vector fields on $M$.

• Metrizable Banach manifolds have the homotopy type of CW-complexes, as shown by Palais.

[Remark: When $M$ is closed, the spaces of $C^k$ diffeomorphisms of $M$ (for $0 < k \leq\infty$) are all homotopy equivalent to each other via the natural inclusions. This can be shown by embedding $M$ smoothly in $\RR^N$, and then using smoothing operators defined by taking convolution with a mollifier.]

When we allow $M$ to not be compact, there are several common topologies on $\Diff(M)$. I am interested in the compact-open (or weak) $C^1$ topology.

Questions: Is it known whether the space $\Diff(M)$ of $C^1$ diffeomorphisms with the compact-open $C^1$-topology is homotopy equivalent to a CW-complex when $M$ is a smooth manifold without boundary? Is there a known counter-example? Are there particular cases where the answer is known, for example if $M$ is the interior of a compact manifold?

Feel free to use instead $C^k$ diffeomorphisms and/or the compact-open $C^k$ topology for any $0 < k \leq\infty$.

I would also be interested in hearing about any known results related to this question: e.g. for spaces of embeddings of manifolds in the compact-open/weak topology when the source is not the interior of a compact manifold.