I wonder which closed orientable 3-manifolds can be embedded in $\mathbb R^4$ and which in $\mathbb R^5$. Is there a way to determine whether given closed 3-manifold, obtained, say by Dehn surgery on knot, can be embedded into $R^4$ ?

Is the answer known for spherical 3-manifolds (finite fundamental group) ?

I am mainly interested in topological properties of manifolds. The known answers for PL or smooth embedding are of value as well for me. I appreciate any kind of answer. I am not interested in such peculiarities as exotic $R^4$.

  • 2
    $\begingroup$ They all embed in $R^5$ (C.T.C. Wall, BAMS 1956), but the question of embedding in $R^4$ seems a hard one. See openproblemgarden.org/op/… $\endgroup$ – BS. Feb 29 '16 at 14:12
  • 4
    $\begingroup$ related question: mathoverflow.net/q/219313/1345 see also arxiv.org/abs/0810.2346 $\endgroup$ – Ian Agol Feb 29 '16 at 14:48
  • $\begingroup$ Do you mean smooth or PL embeddings, or just topological embeddings? $\endgroup$ – Daniele Zuddas Feb 29 '16 at 15:11
  • $\begingroup$ This latter question is maybe a question to be asked on math.stackexchange, rather than in a comment here. $\endgroup$ – Marco Golla Feb 29 '16 at 17:26
  • 3
    $\begingroup$ Note that every closed orientable 3-manifold immerses into $\mathbb{R}^4$. This follows from Hirsch-Smale. So there are no "easy" obstructions (e.g. characteristic classes) to embedding. $\endgroup$ – Qiaochu Yuan Feb 29 '16 at 18:40

There is an analogy to surfaces in a sense. For 3-manifolds that fibre over surfaces there is a complete answer. For a variety of Seifert-fibred manifolds there are complete answers -- but not all. For example, Seifert-fibred homology spheres are still problematic. The preprint that Ian linked to in his comments has much more results of this kind in it.

At present, in summary:

1) We likely do not have a complete set of invariants that obstruct embedding into $\mathbb R^4$.

2) We appear to be far from knowing all the "natural" constructions of embeddings of 3-manifolds into $\mathbb R^4$ for the manifolds that are known to embed.

It is quite possible there are elements of formal logic obstructing both 1 and 2.

For example, if a compact boundaryless connected 3-manifold embeds in $S^4$ it separates it into two components. It is possible that one or even both of these components has a fundamental group with an unsolvable word problem. This would restrict the kinds of techniques one could use for creating obstructions in (1).

edit: I see Agol and Freedman's paper on this topic as connected to this last concern. 2-manifolds in $S^3$ have the Fox re-embedding theorem. So you could hope for some nice re-embedding theorems for $3$-manifolds in $S^4$. You shouldn't expect too nice a re-embedding theorem in $S^4$, since the tool that makes Fox's theorem work is Dehn's lemma, and the analogies to Dehn's lemma in 4-manifold theory are generally not true.

  • $\begingroup$ Thank you very much for the answers ! I am reading your paper now suggested by Ian Agol. At my present knowledge it is a mistery why Poincare homology sphere has tame topological embedding in $S^4$ but it does not have smooth embedding in $S^4$. $\endgroup$ – Marek Mitros Mar 3 '16 at 14:06
  • 1
    $\begingroup$ @MarekMitros: you are not alone. In principle there is a semi-constructive embedding but it is not easy to visualize. This is a result of Mike Freedman's. The embedding construction involves a type of infinite handle adjunctions. So the fact that you have a tubular neighbourhood is not obvious at all. $\endgroup$ – Ryan Budney Mar 4 '16 at 22:37

Results on the subject seem spotty, but the last of them seems to be:

MR3271270 Reviewed 
Donald, Andrew(4-GLAS-SMS)
Embedding Seifert manifolds in S4. (English summary) 
Trans. Amer. Math. Soc. 367 (2015), no. 1, 559–595. 
  • $\begingroup$ @ScottMorrison I tried the citation thing, it generates the citation, but seems to fail to insert it - am I supposed to cut and paste? $\endgroup$ – Igor Rivin Feb 29 '16 at 14:07
  • $\begingroup$ Scott Morrison can't be notified this way since he did not participate in this thread. However, I'll alert him to your comment. $\endgroup$ – Todd Trimble Mar 1 '16 at 11:34
  • $\begingroup$ After you search, click on your chosen result, then click the "Insert citation" button down in the bottom right of the screen. $\endgroup$ – Scott Morrison Mar 1 '16 at 11:50
  • $\begingroup$ @ScottMorrison I could not find such a button, probably a browser issue... $\endgroup$ – Igor Rivin Mar 1 '16 at 14:27
  • $\begingroup$ Let's sort this out via email. $\endgroup$ – Scott Morrison Mar 1 '16 at 22:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.