Is it possible to improve the Whitney embedding theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:05:13Z http://mathoverflow.net/feeds/question/57549 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57549/is-it-possible-to-improve-the-whitney-embedding-theorem Is it possible to improve the Whitney embedding theorem? Ben McMillan 2011-03-06T08:38:17Z 2011-03-06T23:26:09Z <p><strong>Edited to fix the example, as per Zack's suggestion.</strong> </p> <p>Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to assume that all manifolds are 2nd countable and Hausdorff. Furthermore, let's say that our manifolds are connected and closed.</p> <p>The Whitney embedding theorem states that any smooth \$n\$-manifold may be smoothly embedded into \$\mathbb{R}^{2n}\$. If we consider embeddings into more general \$k\$-dimensional manifolds, is it possible find a '\$n\$-universal' manifold of dimension less than \$2n\$?</p> <p>For example, a non-orientable 2-manifold cannot be embedded into \$\mathbb{R}^3\$, demonstrating the sharpness of the Whitney embedding theorem. </p> <p>However, there are 3-manifolds into which we can embed any surface, such as \$M = \mathbb{RP}^3 \sharp \mathbb{RP}^3\$. Indeed, by the classification of surfaces we know that any surface may be decomposed as a connected sum of copies of \$\mathbb{RP}^2\$ and tori. In fact, by the monoid structure of closed surfaces under connected sums we may take this sum to have at most 2 copies of the projective plane. Now, embed 2 disjoint copies of the projective plane into \$M\$ and arbitrarily many copies of the torus. Taking the connected sum of these we see that any closed surface is embeddable into \$M\$.</p> <p>Can we do something similar in higher dimensions?</p> http://mathoverflow.net/questions/57549/is-it-possible-to-improve-the-whitney-embedding-theorem/57598#57598 Answer by Ryan Budney for Is it possible to improve the Whitney embedding theorem? Ryan Budney 2011-03-06T20:56:49Z 2011-03-06T20:56:49Z <p>Yes, the Whitney theorem can be improved in many cases. For example, C.T.C. Wall proved all 3-manifolds embed in \$\mathbb R^5\$. </p> <p>Precisely what is the optimal minimal-dimensional Euclidean space that all \$n\$-manifolds embed in, I don't know what the answer to that is but Whitney's (strong) embedding theorem is only best-possible for countably-many \$n\$, not for all \$n\$. See Haefliger's work on embeddings -- I believe he noticed many cases where you can improve on Whitney. </p> <p>The suggestion to improve Whitney's theorem that you're giving -- making the target not a Euclidean space but a manifold -- in a sense you're asking for something much weaker than Whitney's theorem. For example, given any \$n\$-manifold, you can take its Cartesian product with \$S^1\$. Take the connect sum of all manifolds obtained this way. It's a giant, non-compact \$(n+1)\$-manifold, and all \$n\$-manifolds embed in it. This isn't so interesting. </p> http://mathoverflow.net/questions/57549/is-it-possible-to-improve-the-whitney-embedding-theorem/57600#57600 Answer by Bruno Martelli for Is it possible to improve the Whitney embedding theorem? Bruno Martelli 2011-03-06T21:16:26Z 2011-03-06T21:16:26Z <p>I assume that all manifolds are closed and connected (embedding in \$\mathbb R^n\$ or in \$S^n\$ is the same), but not necessarily orientable.</p> <p>Concerning dimension 3, a famous <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&amp;pg1=CNO&amp;s1=175139&amp;loc=fromrevtext" rel="nofollow">theorem of Wall</a> states that every closed 3-manifold embeds in \$S^5\$.</p> <p>It is not possible to improve this result. A <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.ojm/1200783230&amp;page=record" rel="nofollow">theorem of Shiomi</a> shows that there is no closed 4-manifold which contains every possible closed 3-manifold.</p> <p>The question in dimension 4 seems more complicate. Since \$4=2^2\$, this is one of the dimensions where real projective plane \$\mathbb R\mathbb P^4\$ embeds in \$S^8\$ and not in \$S^7\$. Every <i>orientable</i> 4-manifold embeds <i> topologically </i> in \$S^7\$ by a <a href="http://www.ams.org/mathscinet/search/publdoc.html?amp=&amp;loc=refcit&amp;refcit=1286925&amp;vfpref=html&amp;r=3&amp;mx-pid=1923991" rel="nofollow">theorem of Fang and Fuquan</a>. </p> http://mathoverflow.net/questions/57549/is-it-possible-to-improve-the-whitney-embedding-theorem/57614#57614 Answer by John Klein for Is it possible to improve the Whitney embedding theorem? John Klein 2011-03-06T23:12:18Z 2011-03-06T23:26:09Z <p>If we restrict our interest to manifolds which are \$k\$-connected, then Wall proved that any \$n\$-manifold (closed) \$M\$ admits a locally flat PL embedding in \$\Bbb R^{2n-k}\$, thereby improving on Whitney by \$k\$ dimensions. If in addition we assume the <em>metastable range</em> condition \$2k &lt; n\$, then we can even take the embedding to be smooth. The latter theorem was also known to Haefliger and Hirsch and is historically earlier.</p> <p>One further thing worth mentioning: the Hirsch Conjecture says that a stably parallelizable \$n\$-manifold is supposed to embed in \$\Bbb R^m\$, where \$m = \lceil (3/2)n\rceil \$. The conjecture is still open. Partial results are known: for example it's true when the manifold is \$[n/4]\$-connected.</p>