# Does the self-homeomorphism group of a finite CW complex have CW homotopy type?

Let $$X$$ be a finite CW complex and form the group $$\mathcal{H}(X)$$ of self-homeomorphisms $$X\xrightarrow{\cong}X$$, furnishing it with the compact-open topology. Under the assumptions on our space $$\mathcal{H}(X)$$ is a topological group. My very simple question is the following:

If $$X$$ is a finite CW complex, then does $$\mathcal{H}(X)$$ have CW homotopy type?

I couldn't find this question addressed in the literature. To my knowledge it is an open question as to whether $$\mathcal{H}(X)$$ is an ANR, even when $$X$$ is a (smooth) compact manifold (although it is in case that $$\dim X=1,2$$). Note however that I am asking for something much weaker.

• Just to be clear, are you asking about the homeomorphisms of the underlying topological space, or are you talking about homeomorphisms of the cell complex -- presumably this would be the subgroup of cellular homeomorphisms? – Ryan Budney Mar 17 at 18:14
• @RyanBudney just the homeomorphisms of the underlying space. The question could be posed more generally, but finite CW complexes tend to be the things I end up caring about, and since they tend to have nice topological properties, it may make an answer more approachable. – Tyrone Mar 17 at 18:49
• I guess there should be a way to "restrict" CW homotopy type of $Map(X,X)$ to $\mathcal{H}(X)$. – user43326 Mar 17 at 20:25
• I'm guessing it's false without some regularity hypothesis on $X$. – Tom Goodwillie Mar 18 at 2:29