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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?

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Maybe you will find this useful (for $n=3$): ams.org/journals/proc/1960-011-02/S0002-9939-1960-0112128-9/… –  Michele Triestino Sep 20 '10 at 17:22
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More generally, what is known about when the group of (orientation preserving) homeomorphisms of an (oriented) smooth manifold deformation retracts onto the group of (orientation preserving) diffeomorphisms. –  Dick Palais Sep 20 '10 at 17:36
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Aren't there general theorems implying that Homeo+(S^n) is a simple group? As the connected component of the identity is a normal subgroup, it would answer the question. –  Maxime Bourrigan Sep 20 '10 at 19:14
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In many cases the diffeomorphism group is "bigger". For example, the space of self-homeomorphisms of the closed $n$ disk fixing the boundary pointwise is contractible, while for large values of $n$ the analogous space of diffeomorphisms has plenty of nontrivial rational homotopy, detected by algebraic $K$-theory. –  Tom Goodwillie Sep 20 '10 at 19:19
    
Couple of days ago there was a nice paper in arxiv on a related topic: Decomposing diffeomorphisms of the sphere arxiv.org/abs/1009.3905 –  Dmitri Sep 25 '10 at 17:35

1 Answer 1

up vote 18 down vote accepted

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.

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, hence the conclusion.

PS : in fact, the detailed story is somewhat complicated. What was known relatively early (due to R. D. Anderson, G. M. Fisher 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 and Edwards-Kirby, 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 R. Brown and H. Gluck 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 (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 by Edwards.

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