Extending diffeomorphisms of Riemannian surfaces to the ambient space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:42:28Z http://mathoverflow.net/feeds/question/37940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37940/extending-diffeomorphisms-of-riemannian-surfaces-to-the-ambient-space Extending diffeomorphisms of Riemannian surfaces to the ambient space William 2010-09-07T01:45:24Z 2010-09-07T07:42:53Z <p>Question 1: Given a smooth Riemannian surface <code>$M$</code> in <code>$R^3$</code> (i.e., a smooth Riemannian 2-manifold embedded in <code>$R^3$</code>) and a diffeomorphism <code>$f: M\rightarrow M$</code> of class <code>$C^{k\geq 2}$</code>, does <code>$f$</code> admit a smooth extension <code>$\tilde{f}$</code> to all of <code>$R^3$</code>? If not always, then are there sufficient conditions?</p> <p>Question 2: If the answer to Q1 is affirmative, then given two diffeomorphisms <code>$f,g: M\rightarrow M$</code> of class <code>$C^{k\geq 2}$</code> which are close in the <code>$C^2$</code>-topology, can we find extensions <code>$\tilde{f}, \tilde{g}$</code> which are also close in the <code>$C^2$</code>-topology?</p> <p>Edit 1: I should add that <code>$M$</code> carries the induced metric (from <code>$R^3$</code>).</p> <p>Edit 2: We can ask a more general question. Say <code>$M$</code> is a smooth Riemannian <code>$m$</code>-manifold. Embed <code>$M$</code> in <code>$R^N$</code> isometrically. Say <code>$f: M\rightarrow M$</code> is a diffeomorphism of class <code>$C^k$</code>. Can we extend <code>$f$</code> smoothly to <code>$R^N$</code>?</p> http://mathoverflow.net/questions/37940/extending-diffeomorphisms-of-riemannian-surfaces-to-the-ambient-space/37941#37941 Answer by Ryan Budney for Extending diffeomorphisms of Riemannian surfaces to the ambient space Ryan Budney 2010-09-07T02:22:37Z 2010-09-07T02:52:56Z <p>Q1: Definately not always. More like "almost never". If the automorphism extends to $\mathbb R^3$, then the bundle $S^1 \ltimes_f M$ would embed in $S^4$. $S^1 \ltimes_f M$ is the bundle over $S^1$ with fiber $M$ and monodromy $f$. The most-commonly used obstructions to embedding in this case are things like the Alexander polynomial, and Milnor signatures. </p> <p>I don't see where the metric on $M$ plays a role for this. </p> <p>If you want to see automorphisms that extend (and do not extend) for your Q1, take a look at my <a href="http://front.math.ucdavis.edu/0810.2346" rel="nofollow">arXiv paper</a>. You'll also find some references to several Jonathan Hillman papers that explore such obstructions. </p> <p>In the case that your surface is unknotted -- bounding handlebodies on both sides (thinking of $M \subset S^3$) then the automorphisms of $M$ that extend in this case are well-known. They're called the mapping class group of a Heegaard splitting of $S^3$. It's an infinite group. Generators are known for it (if I recall, they're the automorphisms induced by handle slides) but off the top of my head I'm not sure how much is known about the structure of the group. Do a little Googling on "mapping class group of a Heegaard splitting of S^3" and you should start finding relevant material. </p> <p>To respond to your 2nd edit, if the co-dimension is high enough all automorphisms extend. This is a theorem of Hassler Whitney's. The basic idea is this, let $f : M \to M$ be an automorphism. Let $i : M \to \mathbb R^k$ be any embedding. So you have two embeddings $i \circ f$ and $i$ of $M$ in $\mathbb R^k$. Any two maps $M \to \mathbb R^k$ are isotopic provided the co-dimension is large enough $k \geq 2m+3$ suffices, for example. So isotope your standard inclusion to the one pre-composed with $f$. The Isotopy Extension Theorem gives you the result. </p> <p>For example, if $\Sigma$ is a Heegaard splitting / the surface is unknotted, $\Sigma \subset \mathbb R^3$ (or $\subset \mathbb S^3$) and you have an automorphism $f : \Sigma \to \Sigma$ a neccessary and sufficient condition for $f$ to extend to $\mathbb R^3$ (or a side-preserving automorphism of the pair $(S^3,\Sigma)$ in the $S^3$ case) is that if $C \subset \Sigma$ is a curve that bounds a disc on either the inside or outside of $\Sigma$ respectively, then $f(C)$ bounds a disc on the inside or outside of $\Sigma$ respectively (here I'm using inside/outside re the Jordan-Brouwer separation theorem). Since the fundamental group of the complement is just a free product of infinite cyclic groups this is something that can be checked rather easily provided you know the map $f$ well enough. </p> http://mathoverflow.net/questions/37940/extending-diffeomorphisms-of-riemannian-surfaces-to-the-ambient-space/37956#37956 Answer by Benoît Kloeckner for Extending diffeomorphisms of Riemannian surfaces to the ambient space Benoît Kloeckner 2010-09-07T07:42:53Z 2010-09-07T07:42:53Z <p>If $f$ is the time $1$ of a vector field, then it is easy to extend (in a stable way ragarding the second part of the question) by extending the vector field. IF $f$ is isotopic to identity you should be able to do the same.</p>