connectivity of the group of orientation-preserving homeomorphisms of the sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:47:48Z http://mathoverflow.net/feeds/question/39403 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39403/connectivity-of-the-group-of-orientation-preserving-homeomorphisms-of-the-sphere connectivity of the group of orientation-preserving homeomorphisms of the sphere Keivan Karai 2010-09-20T16:59:22Z 2010-10-03T14:09:37Z <p>In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:</p> <p>Is the group of orientation-preserving homeomorphisms of the $n$-sphere arc-wise connected?</p> <p>According to Sternberg's paper Kneser has shown this to be true for $n=2$. Does anyone know the current status of the problem?</p> http://mathoverflow.net/questions/39403/connectivity-of-the-group-of-orientation-preserving-homeomorphisms-of-the-sphere/39959#39959 Answer by BS for connectivity of the group of orientation-preserving homeomorphisms of the sphere BS 2010-09-25T17:17:23Z 2010-10-03T14:09:37Z <p>This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the <a href="http://en.wikipedia.org/wiki/Annulus_theorem" rel="nofollow">annulus conjecture</a>. </p> <p>The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is <em>stable</em> i.e. a (finite) product of homeomorphisms each of which is the identity on some non-empty open set. </p> <p>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 <a href="http://en.wikipedia.org/wiki/Alexander%2527s_trick" rel="nofollow">Alexander's trick</a>, hence the conclusion.</p> <p><strong>PS</strong> : in fact, the detailed story is somewhat complicated. What was known relatively early (due to <a href="http://www.ams.org/mathscinet-getitem?mr=98145" rel="nofollow">R. D. Anderson</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=117712" rel="nofollow">G. M. Fisher</a> 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.</p> <p>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 <a href="http://www.ams.org/mathscinet-getitem?mr=236948" rel="nofollow">Cernavskii</a> and <a href="http://www.ams.org/mathscinet-getitem?mr=283802" rel="nofollow">Edwards-Kirby</a>, who proved the local contractibilty (hence local arcwise connectedness) of $H(M)$ in any dimension.</p> <p>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.</p> <p>Then <a href="http://www.ams.org/mathscinet-getitem?mr=145497" rel="nofollow">R. Brown and H. Gluck</a> 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).</p> <p>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$_3$.</p> <p>After that came <a href="http://www.ams.org/mathscinet-getitem?mr=242165" rel="nofollow">R. Kirby</a> (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 <code>$n&gt; 4$</code>.</p> <p>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 <a href="http://www.ams.org/mathscinet-getitem?mr=780581" rel="nofollow">survey</a> by Edwards.</p>