What are some of the big open problems in 3-manifold theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:59:17Z http://mathoverflow.net/feeds/question/32292 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory What are some of the big open problems in 3-manifold theory? Paul Siegel 2010-07-17T16:50:05Z 2012-12-08T13:30:10Z <p>From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot of activity. I am not prepared to make a full push to familiarize myself with the literature in the near future, but I am still curious to know what people are working on, what techniques are being developed, etc.</p> <p>So I was hoping people could briefly explain some of the main open questions and programs that are motivating current research on 3-manifolds. I'll let the community decide if this undertaking is too broad, but I'm hoping it is possible to give a rough impression of what is going on. References to survey articles are appreciated, especially if they are accessible to non-experts like myself.</p> <p>It seems like the community wiki designation is appropriate for this question, and the usual rules ought apply.</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/32297#32297 Answer by algori for What are some of the big open problems in 3-manifold theory? algori 2010-07-17T17:17:03Z 2010-07-17T17:17:03Z <p>The volume conjecture. See e.g. H. Murakami's survey <a href="http://arxiv.org/abs/1002.0126" rel="nofollow">http://arxiv.org/abs/1002.0126</a> and references therein.</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/32364#32364 Answer by Daniel Moskovich for What are some of the big open problems in 3-manifold theory? Daniel Moskovich 2010-07-18T14:08:02Z 2010-07-18T14:08:02Z <p>The <a href="http://en.wikipedia.org/wiki/Virtually_fibered_conjecture" rel="nofollow">Virtually Fibred Conjecture</a>, and related problems.<br> For a weaker definition of 3-manifold topology, I think the <a href="http://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture" rel="nofollow">Andrews-Curtis conjecture</a> is a key problem. Also, anything which relates to the classification of non-simply-connected topological 4-manifolds, for instance problems related to knot and link concordance. </p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/32366#32366 Answer by Noah Snyder for What are some of the big open problems in 3-manifold theory? Noah Snyder 2010-07-18T14:29:55Z 2010-07-19T05:35:45Z <p>Rob Kirby has a huge collection of open problems: <a href="http://math.berkeley.edu/~kirby/problems.ps.gz" rel="nofollow">http://math.berkeley.edu/~kirby/problems.ps.gz</a></p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/32391#32391 Answer by algori for What are some of the big open problems in 3-manifold theory? algori 2010-07-18T19:39:34Z 2010-07-18T19:39:34Z <p>If the 3-manifold theory is understood broadly enough, then one should mention the Vassiliev conjecture and in general, the problem of computing the cohomology of the spaces of knots in 3-manifolds. Note that for manifolds of dimension 4 or more this has been completely solved. For an introduction to all this see Vassiliev's ICM 1994 talk (MR1403923) and for more details see his Complements of discriminants of smooth maps (MR1168473).</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/36402#36402 Answer by Sam Nead for What are some of the big open problems in 3-manifold theory? Sam Nead 2010-08-22T21:33:34Z 2010-08-22T21:33:34Z <p>The simple loop conjecture. The statement can be found <a href="http://lanl.arxiv.org/abs/math.GT/0307283" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/36406#36406 Answer by HW for What are some of the big open problems in 3-manifold theory? HW 2010-08-22T22:07:15Z 2010-08-31T23:58:35Z <p>Let $M$ be a finite-volume hyperbolic 3-manifold. (Some of these extend, suitably restated, to larger classes of 3-manifolds. But it follows from Geometrisation that the hyperbolic case is often the most interesting. These are all <strike>trivial or</strike> trivially false in the elliptic case, for example.)</p> <p><strong>The Surface Subgroup Conjecture (SSC).</strong> $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed surface. (Recently proved by Kahn and Markovic.)</p> <p><strong>The Virtually Haken Conjecture (VHC).</strong> $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.</p> <p><strong>Virtually positive first Betti number (VPFB).</strong> $M$ has a finite-sheeted covering space $\widehat{M}$ with $b_1(\widehat{M})\geq 1$.</p> <p><strong>Virtually infinite first Betti number (VIFB).</strong> $M$ has finite-sheeted covering spaces $\widehat{M}_k$ with $b_1(\widehat{M}_k)$ arbitrarily large.</p> <p><strong>Largeness (L).</strong> $\pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.</p> <p><strong>The Virtually Fibred Conjecture (VFC).</strong> $M$ has a finite sheeted cover that is homeomorphic to the mapping torus of a (necessarily pseudo-Anosov) surface automorphism. This is false for graph manifolds. There are fairly easy implications</p> <p>$L\Rightarrow VIFB \Rightarrow VPFB \Rightarrow VHC \Rightarrow SSC$.</p> <p>Also, <em>a fortiori</em>,</p> <p>$VFC\Rightarrow VPFB$.</p> <p>Recently, Daniel Wise announced a proof that $VHC\Rightarrow VFC$. His proof also shows that, if $M$ has an embedded <em>geometrically finite</em> subsurface, then we get $L$ and other nice properties.</p> <p>This list is similar to the one that Agol links to in the comments. Also, I suppose it's exactly what Daniel Moskovich meant by 'The Virtually Fibred Conjecture, and related problems'. I thought some people might be interested in a little more detail.</p> <hr> <p>Paul Siegel asks in comments: 'Would it be correct to guess that the "virtually _ conjecture" problems can be translated into a question about the large scale geometry of the fundamental group?'</p> <p>Certainly, it's true that most of these can be translated into an assertion about how (some finite-index subgroup of) $\pi_1M$ splits as an amalgamated product, HNN extension or, more generally, as a graph of groups. The equivalence uses the Seifert--van Kampen Theorem in one direction, and something like Proposition 2.3.1 of <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=culler&amp;s5=shalen&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=23&amp;mx-pid=683804" rel="nofollow">Culler--Shalen</a> in the other. Rephrased like this, some of the above conjectures turn out as follows.</p> <p><strong>The Virtually Haken Conjecture (VHC).</strong> $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits.</p> <p><strong>Virtually positive first Betti number (VPFB).</strong> $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as an HNN extension.</p> <p><strong>Largeness (L).</strong> $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as a graph of groups with underlying graph of negative Euler characteristic.</p> <p><strong>The Virtually Fibred Conjecture (VFC).</strong> $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits can be written as a semi-direct product</p> <p>$\pi_1(\widehat{M}) \cong K\rtimes\mathbb{Z}$</p> <p>with $K$ finitely generated. (Here we invoke <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=stallings%252C%2520j%2a&amp;s5=&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=41&amp;mx-pid=158375" rel="nofollow">Stallings' theorem</a> that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)</p> <p>I don't think I know a way to rephrase $VIFB$ in terms of splittings of $\pi_1$.</p> <p>Often, when people say 'the large scale geometry of $\pi_1$' they're talking about properties that are invariant under quasi-isometry. I'm really not sure whether these splitting properties (or, more exactly, 'virtually having these splitting properties') are invariant under quasi-isometry. Perhaps something like the work of <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=whyte%252C%2520k%2a&amp;s5=&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=7&amp;mx-pid=1998479" rel="nofollow">Mosher--Sageev--Whyte</a> does the trick?</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/36594#36594 Answer by BMann for What are some of the big open problems in 3-manifold theory? BMann 2010-08-24T22:32:45Z 2010-08-24T22:32:45Z <p>The rank versus Heegaard genus conjecture: It states that given a closed (compact works as well, I think) hyperbolic 3-manifold $M$, the conjecture states that the Heegaard genus of $M$ is equal to the rank of $\pi_1(M)$. It's is relatively easy to see that the Heegaard genus is always greater than or equal to the rank just by looking at the definition of a Heegaard splitting, but the other inequality is not known. </p> <p>For non-hyperbolic 3-manifolds, there are examples where the genus is strictly greater than the rank (I don't have a reference for this off the top of my head).</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/36621#36621 Answer by Ryan Budney for What are some of the big open problems in 3-manifold theory? Ryan Budney 2010-08-25T04:22:59Z 2010-08-25T04:44:36Z <p>I suppose I'm a little contrary but I don't consider the virtual fibering conjecture to really be a big problem in 3-manifold theory. In an earlier era when it might have been an approach to proving geometrization, sure, but nowadays with geometrization a fixture of the landscape, the problem is far less important. Still quite significant, but no longer vital, and I'd rank it well below these problems:</p> <ul> <li><p>Find an algorithmic formalism for the Ricci flow (with surgery). i.e. find a combinatorial formalism for curvature on a manifold and the resulting flow. This should be compatible with means for representing surfaces in the 3-manifold so that surgery can be implemented, for example, a formalism using triangulations of the manifold so that it would be compatible with normal surface theory. Likely you would want a suitable notion of <a href="http://mathoverflow.net/questions/35061/a-pachner-complex-for-triangulated-manifolds" rel="nofollow">Pachner complex</a> to get this formalism off the ground. </p></li> <li><p>Build stronger connections between the geometric perspective on 3-manifolds and other perspectives on 3-manifolds. I would put problems like understanding the properties of the Gordian graph of knots in here. Or the volume conjecture. 4-manifold theory enters the picture here because the question of how geometrization relates to surgery is a big one. Questions like which (rational) homology spheres bound (rational) homology balls, embedding 3-manifolds in 4-manifolds, etc. </p></li> </ul> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/37283#37283 Answer by Bruno Martelli for What are some of the big open problems in 3-manifold theory? Bruno Martelli 2010-08-31T17:27:04Z 2010-08-31T18:28:01Z <p>The geometization conjecture shows that every 3-manifold decomposes into geometric pieces. In some sense, the non-hyperbolic pieces are "well-known" since many decades, whereas the hyperbolic pieces are not. Therefore the current research focuses mainly in "understanding" hyperbolic 3-manifolds. Of course, "understanding" is not a well-defined mathematical problem: however, I think that researchers in the field mostly agree that we are still far from reaching this goal. </p> <p>For instance, as opposite to Seifert manifolds, hyperbolic manifolds are not classified in a strict sense: every Seifert manifold has a standard unique "name" which tells many things about its geometry and topology, but hyperbolic manifolds do not have such univoque names. Volumes of hyperbolic manifolds are still poorly understood, and even a simple relationship between manifolds like a "topological covering" is far from being understood, all the conjectures listed by Wilton above show.</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/37393#37393 Answer by Igor Belegradek for What are some of the big open problems in 3-manifold theory? Igor Belegradek 2010-09-01T15:45:40Z 2010-09-01T15:45:40Z <p>Here are two problems on 3-manifold groups (i.e. fundamental groups of compact 3-manifolds) that I find important.</p> <p>a. Are 3-manifold groups linear? </p> <p>Comments: A group is called <i> linear</i> if it is isomorphic to a subgroup of $GL(n,\mathbb C)$. One can also ask this over other fields but let's focus on $\mathbb C$. Thurston conjectured that 3-manifold groups are linear, because the geometrization implies that they are residually finite (which is weaker than linearity for finitely generated groups). Aschenbrenner-Friedl recently <a href="http://arxiv.org/abs/1004.3619" rel="nofollow">showed</a> that 3-manifold groups are virtually residually $p$ for all but finitely many $p$'s, which again is known for fg linear groups. </p> <p>b. Is it true that every 3-dimensional Poincare duality group is a 3-manifold group?</p> <p>Comments: This is wide open, but see e.g. <a href="http://front.math.ucdavis.edu/0410.5043" rel="nofollow">this survey</a> of Wall, and <a href="http://www.maths.usyd.edu.au/u/jonh/pdq.pdf" rel="nofollow">this list</a> of questions by Hillmann.</p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/37404#37404 Answer by Agol for What are some of the big open problems in 3-manifold theory? Agol 2010-09-01T17:19:38Z 2010-09-01T17:35:26Z <p>One of the unresolved questions about 3-manifolds is the generalized Smale conjecture, which roughly interpreted asks for the homotopy type of the space of diffeomorphisms of a 3-manifold. Smale originally conjectured that $Diff(S^3)\simeq O(4)$, and this was <a href="http://www.jstor.org/stable/200703" rel="nofollow">proven by Hatcher</a>. He also worked out the homotopy type of diffeomorphisms of Haken 3-manifolds. Another interpretation of Smale's question is that the space of round (constant sectional curvature $=1$) metrics on $S^3$ is contractible. Gabai proved the analogous statement that <a href="http://projecteuclid.org/euclid.jdg/1090348284" rel="nofollow">the space of hyperbolic metrics on a hyperbolic 3-manifold is contractible</a>, and recently <a href="http://front.math.ucdavis.edu/1005.5061" rel="nofollow">McCullough and Soma</a> have dealt many small (non-Haken) Seifert-fibered spaces. However, the case of <strong>the generalized Smale conjecture for elliptic manifolds is still open</strong> (see however the work of <a href="http://www.math.ou.edu/~dmccullough/research/preprints.html" rel="nofollow">Hong et. al.</a>). I think this is an important open question, and it would be useful to have a unified proof of these results (in particular, Gabai's results makes use of a <a href="http://front.math.ucdavis.edu/9609.5207" rel="nofollow">computer-aided proof</a> of the existence of "non-coalescable insulator families"). </p> <p>One possible approach is to try to prove that the space of metrics is contractible (on a constant curvature manifold) by showing that all the homotopy groups vanish (it is known to be of the homotopy type of a CW-complex, so this suffices). This was the approach that Gabai took. You can fill in a sphere of constant curvature metrics with a ball of Riemannian metrics, since the space of Riemannian metrics is convex. Then you could try to "flow" towards a ball of constant curvature metrics using Ricci flow (which would stay fixed on the boundary of the ball). The issue is that under Ricci flow, singularities may occur. However, what I hope is that <a href="http://mathoverflow.net/questions/35584/does-ricci-flow-with-surgery-come-from-sections-of-a-smooth-riemannian-manifold/35586#35586" rel="nofollow">some sort of canonical Ricci-flow with surgery</a> may be used to fill in the sphere with a ball of constant curvature metrics. Thus, I see it as an important question for 3-manifold topology to <strong>obtain an understanding of a version of Ricci flow-with-surgery and Perelman's proof of geometrization for families of Riemannian metrics.</strong> This approach for more general Seifert fibered spaces would be trickier, since one would probably have to get a very good idea of how the collapsing occurs at infinite time under Ricci flow, and prove finiteness of surgeries. </p> http://mathoverflow.net/questions/32292/what-are-some-of-the-big-open-problems-in-3-manifold-theory/89678#89678 Answer by Lee Mosher for What are some of the big open problems in 3-manifold theory? Lee Mosher 2012-02-27T16:58:47Z 2012-02-27T16:58:47Z <p>Cannon's Conjecture: Every finitely generated word hyperbolic group with Gromov boundary $S^2$ has a finite normal subgroup whose quotient is the fundamental group of a closed hyperbolic 3-orbifold. </p>