My question is easy to formulate:

**What is known about the homotopy groups of Homeo($\mathbb{R}^n$)?
Especially, what is its fundamental group?** (A guess would be $\mathbb{Z}$ for $n=2$ and $\mathbb{Z}/2$ for large $n$.)

- It is well-known that it has two components, determined by the orientation behaviour.
- Stably a lot is known: One can use either Sullivans results about the homotopy type of $G/TOP$ and that the homotopy groups of $G$ are the stable homotopy groups of spheres in degrees $i\geq 1$. Also, $TOP/PL$ is famously 2-connected, being a $K(\mathbb{Z}/2,3)$. If I understand it correctly, in [Brumfiel, On the homotopy groups of $BPL$ and $PL/O$] it is also mentioned in the introduction that $PL/O$ is at least $2$-connected(*). This would show $\pi_1(TOP) = \mathbb{Z}/2 $.
- Maybe there are results on the connectivity of the map Homeo($\mathbb{R}^n$)$\rightarrow TOP$?

(*) In the paper it is mentioned that $\pi_k(PL/O)$ is isomorphic to the group of concordance classes of smoothings of the $k$-sphere. I guess low dimensional smoothing theory implies that any two smooth structures on $S^k$ ($k\leq 2$) are concordant, since a structure on $S^k\times \{0,1\}$ extends to a structure on $S^k \times I$.