**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. 



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> **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?
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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.


**Edit:** Allen Hatcher gave a very nice answer to my question. Afterwards, I also posted an answer very similar to Allen's, which I was writing when Allen posted his. A pertinent question still remains: (1) Does the result hold for the interior of a compact manifold? Here is a perhaps less pertinent question: (2) For $M$ without boundary, do the path components of $\Diff(M)$ have the homotopy type of a CW-complex?



Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one noncompact end. I will describe an infinite sequence of diffeomorphisms $f\_n:M\to M$ converging to the identity in the compact-open topology and all lying in different path-components of ${\rm Diff}(M)$. Assuming this, suppose $\phi:{\rm Diff}(M) \to X$ is a homotopy equivalence with $X$ a CW complex. The infinite sequence $f\_n$ together with its limit forms a compact set in ${\rm Diff}(M)$, so its image under $\phi$ would be compact and hence would lie in a finite subcomplex of $X$, meeting only finitely many components of $X$. Thus $\phi$ would not induce a bijection on path-components, a contradiction.


To construct $f\_n$, start with an infinite sequence of disjoint simple closed curves $c\_n$ in $M$ marching out to infinity, and let $f\_n$ be a Dehn twist along $c\_n$. The $f\_n$'s converge to the identity in the compact-open topology since the $c\_n$'s approach infinity. We can choose the $c\_n$'s so that they represent distinct elements in a basis for $H\_1(M)$ and then the $f\_n$'s will induce distinct automorphisms of $H\_1(M)$. If two different $f\_n$'s were in the same path-component of ${\rm Diff}(M)$ they would have to induce the same automorphism of $H\_1(M)$ since any path joining them would restrict to an isotopy of any simple closed curve in $M$ (see the next paragraph below) and a basis for $H\_1(M)$ is represented by simple closed curves.


If $g\_t$ is a path in ${\rm Diff}(M)$ then the images $g\_t(c)$ of any simple closed curve $c$ vary by isotopy since this is true as $t$ varies over a small neighborhood of a given $t\_0$, so since the $t$-interval $[0,1]$ is compact, a finite number of these neighborhoods cover $I$ and the claim follows.


Remark: The $f\_n$'s were chosen to be Dehn twists just for convenience. Many other choices of diffeomorphisms would work just as well. One can easily see how to generalize to higher dimensions.


