Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

Question

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

The group $Diff_c(M)$ of compactly supported diffeomorphisms is a nuclear LF space: http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf With this topology does it have the homotopy type of a CW complex?

Answer 1

I'll explain the claim left in the comments - namely that any paracompact manifold modeled on a strict LF space has CW homotopy type. The notion of an LF space is not completely standard across the literature, so to be clear, I will understand a strict LF space to be a locally convex TVS $E$ which is obtained as the inductive limit (in the category of locally convex TVS's) of an increasing sequence of continuous, linear, proper, closed embeddings of Fréchet spaces $$E_1\subseteq E_2\subseteq E_3\subseteq\dots$$ Since the linked monograph doesn't seem to contain any discussion of $Diff_c(M)$, I'm afraid that I really don't know if this is applicable to whatever structure on it that you have in mind.

In the sequel I will make use of spaces which are stratifiable in the sense of Borges [2]. Every metric space is stratifiable, and every stratifiable space is a paracompact, perfectly normal Hausdorff $\sigma$-space. The property of stratifiability is equivalent to the $M_2$ (equivalently $M_3$) property of Ceder [4], and this makes the following easy to prove.

Proposition: A paracompact space which is locally stratifiable is stratifiable. $\quad\blacksquare$

A space $Y$ is an AE(stratifiable) (ANE(stratifiable)) if for any stratifiable space $X$ and any closed subspace $A\subseteq X$, any map $f:A\rightarrow Y$ has an extension to $X$ (to a neighbourhood of $A$ in $X$). A stratifiable space $X$ is said to be an an AR(stratifiable) (ANR(stratifiable)) if whenever it is embedded in a stratifiable space as a closed subset, then it is a retract (neighbourhood retract) there.

Proposition: The following statements hold.

1. AR(stratifiable) $\Rightarrow$ ANR(stratifiable) and AE(stratifiable) $\Rightarrow$ ANE(stratifiable).
2. The property of being ANE(stratifiable) is open-hereditary.
3. A stratifiable space is an AR(stratifiable) (ANR(stratifiable)) if and only if it is an AE(stratifiable) (ANR(stratifiable)).
4. A paracompact space which is locally an ANE(stratifiable) is an ANE(stratifiable). $\quad\blacksquare$

A metric space is an ANE(stratifiable) if and only if it is an ANR(metric). Every CW complex is stratifiable and an ANR(stratifiable). The relevance of this class of spaces to your question is the following.

Theorem: The following statements about a space $X$ are equivalent.

1. $X$ has the homotopy type of a CW complex.
2. $X$ has the homotopy type of a simplicial complex.
3. $X$ has the homotopy type of a metric space which is an ANR(metric).
4. $X$ has the homotopy type of a stratifiable space which is an ANR(stratifiable). $\quad\blacksquare$

The equivalence of the first three bullet points is well known. The fact that they are equivalent to the fourth is a corollary of Théorème 1.5 in Cauty [3].

The following extends the classical Dugundji Theorem, which states that every locally convex TVS is an AE(metric).

Theorem (Borges [1]): Every locally convex TVS is an AE(stratifiable). $\quad\blacksquare$

Now, this theorem covers strict LF spaces, which are therefore AE(stratifiable). Strict LF spaces are never metrisable, but there is the following more general result of Shkarin, which implies their stratifiability.

Theorem (Shkarin [5, Co.5]): A strict inductive limit of a sequence of metrisable, locally convex TVS's is stratifiable. $\quad\blacksquare$

Combining the last two results yields the following.

Corollary: A strict inductive limit of a sequence of metrisable, locally convex TVS's is an AR(stratifiable). $\quad\blacksquare$

I want to clear that Shkarin defines a strict inductive limit in the same way that I have previously. Namely, as the inductive limit of a sequence of closed, continuous, linear inclusions.

Corollary: Any paracompact manifold modeled on a strict LF space has the homotopy type of a CW complex.

Proof: Any manifold $X$ modeled on TVS which is the strict inductive limit of a sequence of metrisable, locally convex TVS's is locally stratifiable and locally an ANE(stratifiable). If $X$ is paracompact, then it is a stratifiable ANE(stratifiable), and hence an ANR(stratifiable). It follows from Cauty's Theorem that $X$ has the homotopy type of a CW complex. $\quad\blacksquare$

Remarks

• We have openly established something more general than claimed.
• A stratifiable spaces is separable if and only if it is Lindelöf. Hence if $X$ in the last statement is separable, then it is homotopy equivalent to a countable CW complex.

References:

[1]. C. Borges, A Study of Absolute Extensor Spaces, Pacific J. Math. 31(3) (1969) 609-617.

[2]. C. Borges, On stratifiable spaces, Pacific J. Math. 17(1) (1966) 1-16.

[3] R. Cauty, Rétractions dans Les espaces dtratifiables, Bulletin de la S.M.F., tome 102 (1974), 129-149.

[4] J. Ceder, Some generalizations of metric spaces, Pacific J. Math. 11(1) (1961) 105-125 .

[5]. S. Shkarin, On stratifiable locally convex spaces, Russian J. Math. Phys., 6 (1999), 435-460.