How transitive are the actions of symplectomorphism groups ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:56:32Z http://mathoverflow.net/feeds/question/61994 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups How transitive are the actions of symplectomorphism groups ? Somnath Basu 2011-04-17T04:02:44Z 2012-11-30T08:15:26Z <p>This question is motivated by the classical fact from differential geometry : </p> <p><em>Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ acts transtively on the configuration space of $n$-points in $M$ or equivalently it acts $n$-transitively on $M$.</em></p> <p>As I recall, it is known that the symplectomorphism group $(M,\omega)$ acts transitively on $M$, which is assumed to be symplectic. My question is then the following :</p> <p><em>Let $(M,\omega)$ be a symplectic manifold.<br> (i) When does $\textrm{Symp}(M,\omega)$ act $n$-transtively for $n\geq 2$ ?<br> (ii) If the answer above is NOT ALWAYS then what is known ?</em></p> <p>As some background, the usual way one proves (rather the only way I know how to prove this) the first fact is by showing the following :<br> (i) for two sets of distinct $n$ points in $M$ given by $\{p_1,\ldots,p_n\}$ and $\{q_1,\ldots,q_n\}$ which are close, we find disjoint disks $D_i$'s containing $p_i,q_i$. This requires dimension at least $2$. Use some diffeomorphism of $D_i$ that is smoothly identity at the boundary and looks like a rotation inside $D_i$ that swaps $p_i$ and $q_i$.<br> (ii) Define the natural equivalence relation on $n$-tuples and observe that the configuration space of $n$-points in $M$. By (ii) each equivalence class is open. It is alsoclosed being the complement of open sets. Since the configuration space is connected (this requires dimension at least $2$) this means there is only one equivalence class. </p> <p>Does this idea work in the symplectic setting - perhaps by taking paths $\gamma_i$ from $p_i$ to $q_i$ and getting Hamiltonian vector fields via $\omega(\gamma_i',\cdot)$ ?</p> http://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups/62018#62018 Answer by Giuseppe for How transitive are the actions of symplectomorphism groups ? Giuseppe 2011-04-17T09:56:42Z 2011-04-17T10:49:56Z <p>Dear Somnath Basu, the answer to question is that, for any $k\geq 2$, the k-fold transitivity of the action of $\mathrm{Sympl}(M,\omega)$ on $M$ has only one obstruction, the trivial one, i.e. connectivity of $M$.<br> But has you proposed there is even more.</p> <p>In particular <a href="http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0236961-0/S0002-9947-1969-0236961-0.pdf" rel="nofollow">Theorem A in a paper of W. Boothby</a> says that, </p> <blockquote> <p>given a connected symplectic manifold $(M,\omega)$, for any two sets $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ of disjoint point in $M$, where $n$ is arbitrary natural number, there exists a time dependent hamiltonian vector field $X_t$ of $(M,\omega)$ such that its evolution operator $K^X_{1,0}$ maps $x_i$ to $y_i$ for $i=1,\ldots,n$.</p> </blockquote> http://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups/62140#62140 Answer by Cédric Bounya for How transitive are the actions of symplectomorphism groups ? Cédric Bounya 2011-04-18T15:44:34Z 2011-04-18T22:26:15Z <p>For any finite number of points, the subgroup of Hamiltonian symplectomorphism that fixes some neighbourhood of the points acts transitively on the complement of the points.</p> <p>Choose two points $a, b$ in the complement, and a path avoiding the constrained points joining them. Then you can find a (non-autonomous) Hamiltonian function with support in an arbitrarily small neighborhood of the path whose time-1 flow maps $a$ to $b$. Indeed, the group of Hamiltonian tranformations with support in a connected neighbourhood is transitive on Darboux balls charts, so the orbit of a point is open and closed.</p> <hr> <p>What this argument seems to prove is in fact the statement :</p> <p>For any connected open subset $\Omega$ in a symplectic manifold, the group of Hamiltonian symplectomorphisms with support in $\Omega$ is transitive in $\Omega$.</p> <p>In particular, the group of compactly Hamiltonian symplectomorphisms is $n$-transitive $\forall n$ on any connected symplectic manifold.</p> http://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups/114962#114962 Answer by Peter Michor for How transitive are the actions of symplectomorphism groups ? Peter Michor 2012-11-30T08:15:26Z 2012-11-30T08:15:26Z <p>Very late, let me point out the following paper:</p> <p>Peter W. Michor, Cornelia Vizman: n-transitivity of certain diffeomorphism groups. Acta Math. Univ. Comenianiae 63, 2 (1994),221--225. arXiv:dg-ga/9406005 <a href="http://www.mat.univie.ac.at/~michor/n-trans.pdf" rel="nofollow">(pdf)</a></p> <p>There $n$-transitivity is proved for many groups of diffeomorphisms, in particular also for the groups of real analytic symplectic, or volume preserving, or contact diffeomorphisms.</p>