What are some of the big open problems in 3-manifold theory?

Question

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.

Answer 1

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 (whose 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.

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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.

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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?

Answer 2

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.

Update: A series of works by Bamler, Kleiner and Lott have carried out this program (I don’t claim that it is original to me) and have proved the existence of a canonical Ricci-flow-with-surgery on compact 3-manifolds and used this to prove the generalized Smale conjecture in all remaining cases.

Rather than cite all of the papers that contributed to this program, I’ll note that the final case of the generalized Smale conjecture was settled recently for Nil manifolds. For further references, see the survey paper of Bamler, New developments in Ricci flow with surgery.

Answer 3

The volume conjecture. See e.g. H. Murakami's survey https://arxiv.org/abs/1002.0126 and references therein.

Answer 4

Rob Kirby has a huge collection of open problems: http://math.berkeley.edu/~kirby/problems.ps.gz

Answer 5

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: Here a group is called linear if it is isomorphic to a subgroup of $GL(n,\mathbb C)$ for some $n$. 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 Poincaré 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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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).

Answer 10

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).

Answer 11

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.

Answer 12

Thurston's remaining challenge:

Show that volumes of hyperbolic 3-manifolds are not all rationally related.

THURSTON'S VISION AND THE VIRTUAL FIBERING THEOREM

Is there an algorithm which determines whether two funda- mental groups of 3-manifolds are isomorphic?

DECISION PROBLEMS FOR 3-MANIFOLDS AND THEIR FUNDAMENTAL GROUPS

Answer 13

The simple loop conjecture. The statement can be found here (in the second full paragraph on the first page).

Answer 14

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.