A senseful meaning of 'approximation of manifolds'? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:25:24Z http://mathoverflow.net/feeds/question/49236 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds A senseful meaning of 'approximation of manifolds'? Martin 2010-12-13T09:32:18Z 2011-01-16T07:01:10Z <p>Any continuous function can be uniformly approximated by smooth functions.</p> <p>I would like to have something similar - in what-ever sense - for continuous manifolds.</p> <p>For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.</p> <p>A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, you need a metric on the manifold for the question to make sense, and there are likely to appear difficulties.</p> <p>Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.</p> http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds/49239#49239 Answer by David Feldman for A senseful meaning of 'approximation of manifolds'? David Feldman 2010-12-13T10:15:34Z 2010-12-13T10:21:08Z <p>Here's an example where a sequence of manifolds approximates one given manifold in a definite sense. </p> <p>For $i\in {\Bbb N}$, let $M_i$ be an orientable surface of genus $i$.</p> <p>Let $T$ be the boundary of a tubular neighborhood in ${\Bbb R}^3$ of <code>$$\{ (x,0,0)_{x\ge 0} \} \cup \{ (x,1,0)_{x\ge 0}\}\cup \{ (i,y,0)_{i\in{\Bbb N},y\in[0,1]}\}\ .$$</code></p> <p>Now one can have maps from $T$ to each $M_i$, each map a homeomorphism when restricted to an ever larger open set, in such a way that these open sets exhaust $T$.</p> <p>In general I suppose you'd want a net of manifolds of a fixed dimension and a corresponding directed set of open sets that exhaust the manifold you mean to approximate. You wouldn't even have to have the limit manifold in advance if you had a suitable family of compatible maps connecting the various manifolds in your net.</p> <p>I doubt you can, in any reasonable way, approximate the topology of a compact topological manifold by distinct compact topological manifolds of the same dimension. </p> http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds/49245#49245 Answer by José Figueroa-O'Farrill for A senseful meaning of 'approximation of manifolds'? José Figueroa-O'Farrill 2010-12-13T11:54:21Z 2010-12-13T11:54:21Z <p>I doubt that this is what you are looking for, but I cannot resist mentioning this kind of beautiful approximation. One way to define a (riemannian) manifold is via a commutative version of Connes's spectral triples.</p> <p>As shown by Roggenkamp and Wendland in <a href="http://arxiv.org/abs/hep-th/0308143" rel="nofollow">this beautiful paper</a>, there exist abstract conformal field theories which in a certain limit (typically a large volume limit) give rise to such commutative spectral triples. A expository version of their results can be found <a href="http://arxiv.org/abs/0803.0657" rel="nofollow">in this more recent preprint</a>. </p> http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds/49316#49316 Answer by Sergey Melikhov for A senseful meaning of 'approximation of manifolds'? Sergey Melikhov 2010-12-13T21:12:08Z 2010-12-13T21:39:33Z <blockquote> <p>A senseful meaning of ‘approximation of manifolds’?</p> </blockquote> <p>A simply-looking meaning is the one of Gromov-Hausdorff, as mentioned by Ryan Budney. It has many cool applications, but none that I'm aware of are to topological manifolds.</p> <p>The kind of approximation that one normally uses to prove something about topological manifolds is representing your manifold as an inverse limit of polyhedra (rather than smooth manifolds). Examples of such use are incidentally given below.</p> <p>I don't think that </p> <blockquote> <p>by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\Bbb R^{2n}$.</p> </blockquote> <p>Whitney's embedding theorem applies only to smooth manifolds. To embed a non-triangulable topological manifold in $\Bbb R^{2n+1}$ (note the dimension shift) one "approximates" the manifold by polyhedra and embeds those first (using PL general position); no simpler way is known. This is called the Menger-Nöbeling-Pontryagin embedding theorem (or after various subsets of the 3 authors); a clear proof appears in <em>J. R. Isbell, Embeddings of inverse limits, Ann. of Math. 70 (1959), 73-84</em> and in Isbell's book "Uniform spaces". (Many other books give a less explicit proof based on the Baire category theorem.) </p> <p>If you really want to embed a non-triangulable topological $n$-manifold $M$ in $\Bbb R^{2n}$ (and not just in $\Bbb R^{2n+1}$), this is harder. Earliest results that seem to imply this are in Bryant-Mio and Johnston's 1999 papers in Topology, available from <a href="http://www.maths.ed.ac.uk/~aar/homology" rel="nofollow">Ranicki's website</a>. A more direct (and I think much easier) proof is <a href="http://front.math.ucdavis.edu/0612.5085" rel="nofollow">in this paper</a> (based again on approximation by polyhedra). In wondering about other possible approaches, I don't see how Kirby-Siebenmann could help: they show that $M\times\Bbb R$ is still not a PL manifold, if $n>4$; I don't know if it could be triangulable. </p> http://mathoverflow.net/questions/49236/a-senseful-meaning-of-approximation-of-manifolds/52217#52217 Answer by ansobol for A senseful meaning of 'approximation of manifolds'? ansobol 2011-01-16T07:01:10Z 2011-01-16T07:01:10Z <p>You may also be interested in the geometric measure theory perspective on what is a nonsmooth surface supporting a "kind of differential geometry" (formulation from Frank Morgan's book <em><a href="http://books.google.com/books?id=fM0ISD-uaZoC&amp;pg=PA220&amp;lpg=PA220&amp;dq=morgan+geometric+variational&amp;source=bl&amp;ots=7cWkij76AW&amp;sig=b0-lB6BlxnQfcXf7H8od33esark&amp;hl=en&amp;ei=l5UyTbXHEouWswboudmFCg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBIQ6AEwAA#v=onepage&amp;q=morgan%2520geometric%2520variational&amp;f=false" rel="nofollow">Geometric measure theory: a beginner's guide</a></em>). The relevant notion is that of "rectifiable currents."</p> <p>Specifically regarding convergence of manifolds, have a look e.g. at the paper <em><a href="http://arxiv.org/abs/1006.0411" rel="nofollow">How Riemannian manifolds converge: A survey</a></em> by Christina Sormani (arXiv:1006.0411) and references therein.</p>