In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:


Is the group of orientation-preserving homeomorphisms of the $n$-sphere arc-wise connected?


According to Sternberg's paper Kneser has shown this to be true for $n=2$. Does anyone know the current status of the problem?



connectivity of the group of orientation-preserving homeomorphisms of the sphere
This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the [annulus conjecture](http://en.wikipedia.org/wiki/Annulus_theorem). 


The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is *stable* i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set. 


The same statement is then true for an homeomorphism $h$ of $\mathbb{S}^n$ : first, you can by composition with a (stable) homeomorphism of $\mathbb{S}^n$ assume that $h$ fixes the north pole $p$. Then by SHC $h$ restricted to $\mathbb{S}^n-p\simeq\mathbb{R}^n$ is a composition of homeomorphisms which are the identity on nonempty open sets. But an homeomorphism of $\mathbb{S}^n$ which is the identity on a non-empty open set is isotopic to the identity, by [Alexander's trick](http://en.wikipedia.org/wiki/Alexander%2527s_trick), hence the conclusion.


**PS** : in fact, the detailed story is somewhat complicated. What was known relatively early (due to [R. D. Anderson](http://www.ams.org/mathscinet-getitem?mr=98145), [G. M. Fisher](http://www.ams.org/mathscinet-getitem?mr=117712) around 1960) was that for any topological manifold $M$ (maybe non-compact, but paracompact), the group $H\_c(M)$ of homeomorphisms generated by those compactly supported in domains of topological charts $\mathbb{R}^n\simeq U\subset M$ is the smallest nontrivial normal subgroup of $H(M)$ of all homeomorphisms, and that it is simple. In particular, $H\_c(M)$ is arcwise connected. The proof is ingenious, but not very difficult.


The (much harder) methods and results of geometric topology in dimension $3$ (Bing, Moise,...) then allowed Fisher to prove that for a closed $3$-manifold $M$, $H\_c(M)$ is open in $H(M)$ and thus coincides with the identity component $H\_0(M)$. In particular $H(M)$ is locally arcwise connected, a not at all obvious fact -- later generalized by [Cernavskii](http://www.ams.org/mathscinet-getitem?mr=236948) and [Edwards-Kirby](http://www.ams.org/mathscinet-getitem?mr=283802), who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.


Fisher also considered the group of stable homeomorphisms $H\_s(M)$ (without the name) of a connected $M$ (otherwise the notion is empty). He managed to prove that it coincides with the group of orientation preserving ones $H\_+(M)$ for closed oriented $3$-manifolds $M$ admitting an orientation reversing homeomorphism. For $M=\mathbb{S}^3$, this implies that $H\_+=H\_0=H\_c$ : any orientation preseving homeomorphism of $\mathbb{S}^3$ is isotopic to the identity (note that for all $n$, $H\_s(\mathbb{S}^n)=H\_c(\mathbb{S}^n)$). This is the $n=3$ case of your question.


Then [M. Brown and H. Gluck](http://www.ams.org/mathscinet-getitem?mr=145497) named stable homeomorphisms, and studied stable structures on manifolds.
A puzzling aspect of the notion of stable homeomorphism is that it is very "contagious" : if $h\in H(M)$ coincides with an element $f$ of $H\_s(M)$ on a nonempty open set $U$, then $h$ is in $H\_s(M)$, since $h^{-1}f$ is the identity on $U$. So this is seen locally everywhere, like orientation preservation (to which it was eventually identified).


Brown and Gluck proved that SHC$\_n$ (SHC in dimension $n$) implies the annulus conjecture in dimension $n$ (AC$\_n$), and that AC$\_k$ in all dimensions $k\leq n$ imply SHC$\_n$. But this was still stuck at SHC$\_3$.


After that came [R. Kirby](http://www.ams.org/mathscinet-getitem?mr=242165) (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$\_n$ (hence AC$\_n$) is true in all dimensions $n> 4$.


But the remaining case SHC$\_4$ was only solved by F. Quinn in 1982 (after work of A. Casson and M. Freedman),
who proved AC$\_4$, hence SHC$\_4$ since the $n\leq 3$ cases were known. See the [survey](http://www.ams.org/mathscinet-getitem?mr=780581) by Edwards.


