For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish topological group under the composition operation. Many of the classical results in this area concern the homeomorphism type of $\mathcal H(X)$, for instance:

(0) $\mathcal H(2^\omega)\simeq \mathbb R \setminus \mathbb Q$;

(1) $\mathcal H_\partial([0,1])\simeq \ell^2$ (here $\mathcal H_\partial$ is the space of homeomorphisms which fix the boundary pointwise);

(2) $\mathcal H_\partial([0,1] ^2)\simeq \ell^2$;

(3) $\mathcal H_\partial([0,1]^n)$ is a mystery for $3\leq n<\omega$;

(4) $\mathcal H([0,1]^\omega)\simeq \ell^2$;

(5) $\mathcal H(\text{Sierpinski carpet})$ and $\mathcal H(\text{Menger curve})$ have dimension $1$. It is conjectured that these homeomorphism groups and others are homeomorphic to the $\omega$-power of $\{x\in \ell^2:x_i\notin \mathbb Q\text{ for all }i<\omega\}$;

(6) $\mathcal H(\text{Pseudo-arc})$ contains no continuum. It is unknown whether this group is zero-dimensional, connected, or something in-between.

(7) It is unknown if there is a compact space $X$ with $1<\dim(\mathcal H(X))<\infty$.

**Question 1.** What are some other major results and questions along these lines?

**Question 2.** I suppose the dimension/connectedness of $\mathcal H(X)$ says something about generalized homotopy between homeomorphisms. But specifically why is knowing the homeomorphism type of $\mathcal H(X)$ important/useful? Also are there applications and/or interpretations of $\mathcal H(X)$ which lead to a greater appreciation?

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