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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.

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.

It seems like the community wiki designation is appropriate for this question, and the usual rules ought apply.

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See my talk for some discussion of classification of geometric structures. math.auckland.ac.nz/wiki/… –  Ian Agol Jul 17 '10 at 17:21
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It would be beneficial to the mathematics community if someone knowledgeable updates the relevant Wikipedia entries, which do not seem to be as detailed as the union of the answers to this (good!) question. –  Joseph O'Rourke Aug 24 '10 at 22:50
    
I added tag open-problems-list, it implies poping-up this nice question -- hope this serves the community. –  Alexander Chervov Dec 8 '12 at 13:30
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13 Answers

ADDED (29 May, 2013)

As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaking work of Agol, Kahn--Markovic and Wise. Here's a brief summary of some of the highlights. (Shameless self-promotion: see this survey article for too many further details, including definitions of some of the terms.)

  1. Haglund--Wise define the notion of special (non-positively curved) cube complex. If a closed hyperbolic 3-manifold $M$ is homotopy equivalent to a special cube complex then $M$ satisfies L (largeness, defined below).

  2. Agol proves that if $M$ is homotopy equivalent to a special cube complex then $M$ also satisfies VFC (the Virtually Fibred Conjecture, also defined below).

  3. Kahn--Markovic prove SSC (the Surface Subgroup Conjecture, also defined below), using mixing properties of the geodesic flow. In fact, they construct enough surfaces to show that $M$ is homotopy equivalent to a cube complex.

  4. Wise proves (independently of Kahn--Markovic) that if $M$ contains an embedded, geometrically finite surface then $M$ is special.

  5. Agol uses a very deep theorem of Wise (the Malnormal Special Quotient Theorem) to prove a conjecture (also of Wise), which states that word-hyperbolic fundamental groups of non-positively curved cube complexes are special. All the properties below follow.

It's quite a story, and many other names have gone unmentioned. There were also very important contributions by Sageev (who's thesis initiated the programme of using cube complexes to attack these problems), Groves--Manning, Bergeron--Wise, Hsu--Wise and another very deep paper of Haglund--Wise. To extend these results to the cusped hyperbolic case you need results of Hruska--Wise and Sageev--Wise. Finally, it turns out that similar results hold for all non-positively curved 3-manifolds, a result established by Liu and Przytycki--Wise.


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 trivial or trivially false in the elliptic case, for example.)

The Surface Subgroup Conjecture (SSC). $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. (Recently proved by Kahn and Markovic.)

The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.

Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ with $b_1(\widehat{M})\geq 1$.

Virtually infinite first Betti number (VIFB). $M$ has finite-sheeted covering spaces $\widehat{M}_k$ with $b_1(\widehat{M}_k)$ arbitrarily large.

Largeness (L). $\pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.

The Virtually Fibred Conjecture (VFC). $M$ 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

$L\Rightarrow VIFB \Rightarrow VPFB \Rightarrow VHC \Rightarrow SSC$.

Also, a fortiori,

$VFC\Rightarrow VPFB$.

Recently, Daniel Wise announced a proof that $VHC\Rightarrow VFC$. His proof also shows that, if $M$ has an embedded geometrically finite subsurface, then we get $L$ and other nice properties.

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.


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?'

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 Culler--Shalen in the other. Rephrased like this, some of the above conjectures turn out as follows.

The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits.

Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as an HNN extension.

Largeness (L). $M$ 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.

The Virtually Fibred Conjecture (VFC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits can be written as a semi-direct product

$\pi_1(\widehat{M}) \cong K\rtimes\mathbb{Z}$

with $K$ finitely generated. (Here we invoke Stallings' theorem that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)

I don't think I know a way to rephrase $VIFB$ in terms of splittings of $\pi_1$.

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 Mosher--Sageev--Whyte does the trick?

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At the very least I was interested in a little more detail - thanks for providing it! 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? –  Paul Siegel Aug 24 '10 at 16:54
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The question whether "there splitting properties are quasi-isometry invariant" is answered by a theorem of Richard Schwartz: quasi-isometry is equivalent to commensurability for non-uniform lattices in the isometry group of the hyperbolic 3-space. And all uniform lattices are qi to the hyperbolic space. So there seems to be no qi versions of the above conjectures. –  Igor Belegradek Aug 25 '10 at 15:32
    
Igor, you're right to point that out. But I was wondering more about the other side of the question, namely whether `virtually acting on a tree' is invariant under quasi-isometry. I would guess that it isn't, but that it might be under some fairly mild hypotheses like being $PD_3$. –  HJRW Aug 27 '10 at 20:49
    
Sorry, I should have said $PD_n$. As you point out below, $PD_3$ could be the same as being a 3-manifold group! –  HJRW Sep 1 '10 at 16:12
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Ryan - your comment prompted me to add an update. –  HJRW May 29 '13 at 10:08
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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 proven by Hatcher. 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 the space of hyperbolic metrics on a hyperbolic 3-manifold is contractible, and recently McCullough and Soma have dealt many small (non-Haken) Seifert-fibered spaces. However, the case of the generalized Smale conjecture for elliptic manifolds is still open (see however the work of Hong et. al.). 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 computer-aided proof of the existence of "non-coalescable insulator families").

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 some sort of canonical Ricci-flow with surgery 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 obtain an understanding of a version of Ricci flow-with-surgery and Perelman's proof of geometrization for families of Riemannian metrics. 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.

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Related, there recently was a new proof of Cerf's theorem $\pi_0 Diff(S^3)≃\mathbb Z_2$ put up on the arXiv: front.math.ucdavis.edu/1007.3606 –  Ryan Budney Sep 1 '10 at 17:48
    
Update: I talked a little with John Etnyre about this and he believes this argument can be turned into another proof of Hatcher's theorem, but it sounds like it would be just a re-encoding of Hatcher's proof rather than a completely different proof. –  Ryan Budney May 30 '13 at 15:55
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The volume conjecture. See e.g. H. Murakami's survey http://arxiv.org/abs/1002.0126 and references therein.

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Everytime someone tells to "have a look at [...] and the references therein" I suddenly think of some C++ programs and pointers to pointers to pointers to ... –  Johannes Hahn Jul 18 '10 at 0:54
    
Johannes -- sorry, I know exactly how you feel. What I meant to say was that this is a very good introduction containing the statement of the conjecture and its variants along with the references to the original papers, settled cases etc. –  algori Jul 18 '10 at 3:55
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This is a very nice paper (and quite readable) - thanks! –  Paul Siegel Aug 24 '10 at 16:51
    
Paul -- welcome! –  algori Aug 24 '10 at 22:55
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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:

  • 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 Pachner complex to get this formalism off the ground.

  • 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.

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Ryan, I think your distinction is a little artificial. For instance, I would rate Friedl and Vidussi's proof that a 3-manifold $N$ is fibred if and only if $S^1 \times N$ is symplectic as one of the most attractive examples of 'a strong connection between the geometric perspective on 3-manifolds and other perspectives'. Their proof made crucial use of Agol's work on the Virtual Fibring Conjecture. –  HJRW Aug 25 '10 at 5:23
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Somehow an argument making the point of the interconnectedness of things in a thread which has the intent to artificially single-out individual "big" problems seems contrary to the point. :) My point being largely that 3-manifold theory's future growth should largely be outward to subjects bordering-on 3-manifold theory. And refining geometrization to make it more easily applicable and useful. –  Ryan Budney Aug 25 '10 at 7:04
    
On the other hand, the question didn't ask for opinions as to which problem is 'the biggest'! Actually, I'm interested in your opinion and I don't disagree that (the subject of) 3-manifolds needs to look outwards. Which is why I wanted to point out that some of the problems already mentioned are indeed 'applicable'. –  HJRW Aug 25 '10 at 16:19
    
I'd argue you made the point that some of Agol's work is applicable. VFC may have been part of his interest but it's not VFC that's actually been applied. But that's a small point. –  Ryan Budney Aug 25 '10 at 17:14
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I must respectfully disagree. In light of the Kahn-Markovic theorem, the importance of VFC has been heightened even further. Combining K-M with Thurston's dichotomy for surface subgroups (every surface subgroup of an $H^3$-manifold group is either undistorted or $\pi_1$ of a virtual fiber) VFC is now known to be equivalent to the following: The fundamental group of every closed hyperbolic 3-manifold contains a distorted surface group. I believe it is true (someone correct me if I'm wrong) that the surface subgroups constructed by K-M are all undistorted. –  Lee Mosher Feb 27 '12 at 21:06
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Here are two problems on 3-manifold groups (i.e. fundamental groups of compact 3-manifolds) that I find important.

a. Are 3-manifold groups linear?

Comments: A group is called linear 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 showed that 3-manifold groups are virtually residually $p$ for all but finitely many $p$'s, which again is known for fg linear groups.

b. Is it true that every 3-dimensional Poincare duality group is a 3-manifold group?

Comments: This is wide open, but see e.g. this survey of Wall, and this list of questions by Hillmann.

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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.

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Actually, this is a special case of Wall's conjecture mentioned by Belegradek (PD(3) groups are manifold groups). Of course, Cannon's conjecture might be easier to settle, at least there are some tools available in hyperbolic setting. –  Misha Dec 8 '12 at 16:28
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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.

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).

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This was solved by Tao Li. PDF here: www2.bc.edu/~taoli/rank.pdf . –  HJRW May 29 '13 at 10:21
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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.

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.

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In some sense hyperbolic $3$-manifolds do have names. Their fundamental group functions as a name, since the isomorphism problem is decidable for them. I think in some way the mystery is more the other-way around -- instead of assigning names we want a "phone book". Like: I want a non-repeating list of all hyperbolic 3-manifolds with volume less than 20, how do I generate that list efficently? Similarly, "is $\sqrt{71\sqrt{11}}$ the volume of a hyperbolic 3-manifold?" –  Ryan Budney Aug 31 '10 at 23:51
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The Virtually Fibred Conjecture, and related problems.
For a weaker definition of 3-manifold topology, I think the Andrews-Curtis conjecture 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.

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Can you, please, comment on the precise relation of the VFC to the virtual positive Betti number conjecture? –  Victor Protsak Jul 18 '10 at 16:28
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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).

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FYI, in my papers I reduce the computation of the homology of $Emb(S^1,S^3)$ to a question about certain canonical representations of certain subgroups of the group of hyperbolic isometries of certain hyperbolic link complements (in $S^3$). Here "certain" means the qualifiers are simple to state, just too long for this comment. Your "completely solved" statement needs more qualifiers. Perhaps you're talking about convergence of an appropriate spectral sequence provided the co-dimension of the embeddings is 3 or larger? –  Ryan Budney Aug 25 '10 at 3:58
    
Ryan -- yes, I meant to say the spectral sequences degenerate and one can compute each $E^{pq}_\infty$; these will be finite dimensional and one can in principle construct explicit cohomology classes and evaluate them on each knot. In general (that is if one considers embeddings whose source is not necessarily the circle) codimension 3 does not suffice: one needs the discriminant to be of real codimension $>1$ in order for the spectral sequence to degenerate, so by counting parameters if $n$ is the dimension of the source, the dimension of the target must be at least $2n+2$. –  algori Aug 25 '10 at 5:04
    
.. that is one can compute each $E^{pq}_{\infty}$... (the formula disappears when I press add comment, not sure why) –  algori Aug 25 '10 at 5:08
    
@algori - do you know of a proof that these spectral sequences in dimensions 4 and higher degenerate over the integers? I only know of one for rational cohomology and homotopy, for embeddings of spheres in Euclidean spaces. These degeneration results are by Arone-Lambrechts-Turchin-Volic. Also, explicit cohomology classes are only known (to me) over the reals using de Rham theory (by Cattaneo-Cotta Ramussino-Longoni) and not even over the rationals much less the integers. –  Dev Sinha Sep 1 '10 at 6:59
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Yes, there is a description of the E_1 in those terms - see arxiv.org/pdf/math.AT/0407039.pdf and arxiv.org/pdf/math/0202287.pdf But there is a big difference between convergence and collapse. In both cases "there exists an algorithm" but if all you have is convergence the algorithm is hardly an algorithm. The collapse results of Lambrechts-Turchin-Volic say much more - I suggest you take a look at them if you have an interest in this area. –  Dev Sinha Sep 2 '10 at 4:38
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The simple loop conjecture. The statement can be found here.

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Inspired by the rather lengthy discussion on the low-dimensional topology blog, there's a rather basic question for 3-manifold algorithmics that is still unsolved, as far as I know.

  • If a 3-manifold $M$ admits a complete hyperbolic structure of finite volume, does it admit an ideal triangulation? i.e. the kind of triangulation where the software SnapPy could find its hyperbolic structure.

It would be nice to either understand the hyperbolic manifolds that SnapPy (in its current state) can not deal with. Or if such manifolds do not exist, have a sense for how complicated the triangulation needs to be in order to find the hyperbolic structure.

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Ryan: You probably are aware of this, but, just in case: If you allow degenerate ideal simplices as a part of your triangulation, then the answer is positive and follows immediately from the work of Epstein and Penner. What they construct is a (convex) ideal fundamental polyhedron for an arbitrary complete noncompact hyperbolic n-manifold of finite volume. However, if you do not allow degenerate ideal simplices, then this indeed becomes a known open problem. –  Misha May 30 '13 at 16:55
    
Hi Misha, do you know where this problem is recorded? I've only heard it by word of mouth (via Rubinstein). It would be nice to know a reference. –  Ryan Budney May 30 '13 at 16:58
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Ryan: Look for instance here arxiv.org/pdf/math/9901045v1.pdf or here arxiv.org/pdf/math/0701431v1.pdf –  Misha May 30 '13 at 18:39
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