Anyone who has been following this blog knows that the last few months and years have seen a bewildering amount of progress in what we know about 3-manifolds, and particularly their fundamental groups. Matthias Aschenbrenner, Stefan Friedl and I have recently posted a survey paper on the arXiv, the aim of which is to summarise these recent developments and state, as definitively as possible, what we currently know about 3-manifold groups.

It’s a dauntingly large subject, and there are inevitably many errors and omissions. No doubt we will need to produce an updated version soon. For this reason, we actively encourage comments. We’re particularly keen to hear about mistakes or misattributions, but even if we’ve just omitted your favourite paper (even if you wrote it yourself!) please do drop us a line and let us know. We’d love to hear from you.

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Two comments on the 20 pages I’ve read so far.

On page 10 you state, “Remark. There exists also a version of the Geometrization Theorem for non-

orientable 3-manifolds; we refer to [Bon02, Conjecture 4.1] for the statement.”

This is perhaps a little too breezy. There is a discussion on some of the issues on the webpage,

http://mathoverflow.net/questions/38098/geometrization-for-3-manifolds-that-contain-two-sided-projective-planes

The questions raised in that webpage may have been resolved by Kleiner and Lott’s ArXiv 2011 paper Geometrization of Three Dimensional Orbifolds Via Ricci Flow. But, one needs a little discussion on the role and indeed necessity of the orbifold geometrization theorem for the general orientable/nonorientable Geometrization Theorem. That is definitely NOT found in the reference you give of Bonahon’s article in Daverman’s Handbook of Topology book.

On page 17, your “Theorem 2.1 (Kneser Conjecture) Let N be a compact, orientable 3-mani-

fold with incompressible boundary.”

Doesn’t “Incompressible boundary” here include the case of empty boundary? If so, you should make that explicit since you only defined the term incompressible for surfaces, in your paper.

Anyway, my two cents worth.

Comment by Mayer A. Landau — May 5, 2012 @ 2:56 pm |

Thanks for pointing these out, Mayer.

Comment by Henry Wilton — May 7, 2012 @ 6:45 am |

Maybe I’m missing something obvious, but in your statement of the Hyperbolization Theorem (Theorem 1.13 on page 10), you conclude that the manifold is either Seifert fibered or hyperbolic. Shouldn’t it just be hyperbolic?

Comment by Michael Siler — May 8, 2012 @ 8:28 pm |

Hi Michael – it looks like you’re right. Thanks for pointing this out. We’ve made the easy mistake of mixing up the two definitions of

atoroidal– you can either require that all incompressible tori are homotopic into the boundary, or just the embedded ones. In the latter case, you find that certain Seifert fibred spaces are atoroidal. But we used the former definition.Comment by Henry Wilton — May 9, 2012 @ 12:24 am |

On the prime decomposition theorem, isn’t it better to have a stronger version, namely, to use the version where M=#(X_k)#(Y_j)#(S^1xS^2) where the X_k are spherical and the Y_j are aspherical? The reason I suggest this version is that throughout the paper you implicitly assume that there are only two types of irreducible 3-manifolds, constant positive curvature (i.e. spherical) and aspherical (i.e. \pi_k=0 for k>1). This version of the decomposition theorem makes it clear that there are no other irreducible types in the decomposition. That is, you don’t have anything strange of some kind of mixed type. Of course that follows from some basic algebraic topology, but still, this version makes it explicit.

Comment by Mayer A. Landau — May 31, 2012 @ 11:21 pm |

Mayer – thanks for the comment. The dichotomy you propose follows from Geometrization, which is of course further down the road than the prime decomposition.

Comment by Henry Wilton — June 6, 2012 @ 9:28 am |

Which part follows from Geometrization? Just following Allen Hatcher’s notes we know that a prime 3-manifold is one of three types, finite fundamental group, infinite cyclic, or infinite noncyclic fundamental group. Prior to Geometrization it was known that finite \pi_1 manifolds had homotopy sphere universal covers, infinite cyclic were S^1xS^2, and the last were K(\pi,1) manifolds (using the Hurewicz and Whitehead theorem and the isomorphism of higher homotopy groups for a manifold and its universal cover). So where do you need Geometrization?

Comment by Mayer A. Landau — June 6, 2012 @ 11:33 pm

A 3-manifold is (usually) called spherical if M admits a complete metric of constant curvature one, or equivalently, if M is the quotient of S^3 by a finite discrete subgroup of SO(4).

To conclude that a 3-manifold with finite pi_1 is spherical requires the Elliptization Theorem (which is one of the two components of what’s usually referred to as the Geometrization Theorem)

Comment by Stefan Friedl — June 7, 2012 @ 1:37 am

Hi Stefan, everything you say is true, but … I think misses the point I was trying to make. So, I’ll rephrase.

All of Liu’s paper, and the two Przytycki–Wise papers, and the bulk of your paper, is concerned with irreducible aspherical manifolds. A natural question for a student is, “What about manifolds that are not aspherical?”. The answer supplied by the version of the Prime Decomposition Theorem quoted by me above is that these manifolds are either S^1xS^2 or are covered by the case of manifolds with finite fundamental group. That’s not immediately obvious from the version of the Prime Decomposition Theorem quoted in your paper. Yes, if you want to go further and classify the finite \pi_1 manifolds you need Elliptization and so on, but that gets ahead of my original point.

Comment by Mayer A. Landau — June 8, 2012 @ 12:56 pm |

Thanks again for the suggestion, which isn’t quite what you originally wrote. The next version will include a finer, post-Geometrization, classification statement that I think will satisfy you.

Comment by Henry Wilton — June 8, 2012 @ 1:07 pm

This is really nice! Congratulations on getting it out so fast! I really enjoyed reading it.

My main comments are on presentation rather than on actual content- the main thing is that the “lists” of Section 4 and 6 aren’t very readable. Some items are definitions and some not, why a “non-fibre surface” as opposed to a “fibre surface” is being defined, and idea the definitions are capturing is often missing. I’d recommend rewriting those sections. The other major comment that I had is that some conjectures are motivated by appealing to their pedigree rather than by explaining why they help us classify 3-manifolds and 3-manifold groups- e.g. Section 5.2 and Question 9.9. Also, Theorem 5.4 has no meaning yet, because “virtually compact special” wasn’t defined yet. I’d also recommend a very zoomed-out big-picture section on the extent to which 3-manifold groups are now classified, which would be some sort of “executive summary” of where we stand now (Section 9.1 doesn’t quite do that).

Comment by dmoskovich — June 4, 2012 @ 2:55 am |

Well… for one thing, with the posting on May 31 on the ArXiv of Przytycki and Wise’s paper, “Mixed 3-manifold groups are virtually special”, the only case that remains sort of unknown is that of nontrivial graph manifolds that are not Sol, and that have no spherical and hyperbolic pieces.

Comment by Mayer A. Landau — June 4, 2012 @ 5:51 pm |

Indeed! A new version of the paper, incorporating Przytycki–Wise, is in the pipeline. Following the work of Liu, a more precise statement would be that ‘the only case that remains sort of unknown is that of nontrivial graph manifolds which are not non-positively curved’. We discuss these in Section 9.3.

Comment by Henry Wilton — June 4, 2012 @ 8:45 pm

Many thanks for your comments, Daniel.

I think it’s important to clarify that the paper isn’t really meant to be read from start to finish. (Kudos to you if you succeeded in doing so!) Rather, it’s meant to provide definitive references for as many known facts about 3-manifold groups as possible.

I’m going to respond to some of your comments in turn, since it gives me a good opportunity further clarify what we are and are not trying to do.

the “lists” of Section 4 and 6 aren’t very readable.We’re very conscious that long lists of definitions and references aren’t very readable. On the other hand, long lists of references are the

raison d’etreof the paper, so this unreadability is somewhat inevitable. The diagrams of implications are our attempt to compensate.Some items are definitions and some notWell, some lists are lists of definitions, some are of references for implications and some are of conventions. We try to explain clearly which is which, and hope that there’s no ambiguity. (Of course, sometimes a complicated implication needs a definition.) If you can can give an explicit example of a list whose status is unclear, that would be helpful.

why a “non-fibre surface” as opposed to a “fibre surface” is being definedBecause that’s the term that appears in the diagram.

and idea the definitions are capturing is often missingWith so many definitions to cover, there’s only so much we can do in this direction. Again, if you have any

specificdefinitions for which you think some remarks about intuition would be helpful, we’d be glad to hear them.I’d recommend rewriting those sections.Since compiling those sections was about 90% of the work of writing the paper, I’d say the chances of our re-writing them completely are exactly zero. Of course, in an ideal world they’d be more readable, but as I’ve tried to explain, readability is unfortunately somewhat beside the point.

some conjectures are motivated by appealing to their pedigree rather than by explaining why they help us classify 3-manifolds and 3-manifold groups- e.g. Section 5.2 and Question 9.9.I think that for most people working in 3-manifold topology, Thurston’s questions don’t need too much justification. Someone coming from a different field might well want more explanation, but that’s not really our target audience.

Also, Theorem 5.4 has no meaning yet, because “virtually compact special” wasn’t defined yet.That’s why the next line reads ‘We will give the definition of ‘virtually compact special’ in Section 5.3.’

I’d also recommend a very zoomed-out big-picture section on the extent to which 3-manifold groups are now classified, which would be some sort of “executive summary” of where we stand nowIn a sense, the whole paper is intended as an attempt to do exactly that. The introduction supposed to provide an overview of the basic ingredients. A later version will also include some discussion of decision problems (although the Virtually Compact Special Theorem is relatively unimportant from that point of view—almost everything follows from Geometrization). In particular, we will discuss the Homeomorphism Problem, which is perhaps the most concrete sense in which ‘3-manifold groups are now classified’.

Comment by Henry Wilton — June 4, 2012 @ 8:40 pm |

Henry, don’t you need to still specify that the graph manifold has no spherical components in its JSJ decomposition? Stating that the graph manifold is not non-positively curved seems to only rule out hyperbolic pieces or Sol geometry. The lack of spherical components comes from the prime decomposition theorem, no?

Comment by Mayer A. Landau — June 5, 2012 @ 10:48 pm

Mayer – a graph manifold is by definition irreducible, so its prime decomposition is trivial.

Comment by Henry Wilton — June 6, 2012 @ 7:30 am

Thanks for the reply! I can only comment on how I approach it as a reader. The first 3 sections give a nice overview of some highlights of 3-manifold groups. I sort of expected the paper to continue in that vein. Then, in Sections 4,5, and 6, we get some lists and diagrams of implications, which can’t really be read through, although they’re useful as a reference (or as a guide to literature) if you’re looking for something specific, sort of like a dictionary. In that respect, A4 wasn’t so useful- for example- it gives a reference and not much more. D7 was cryptic for me.

I look forward to a discussion of the Homeomorphism problem!

Comment by dmoskovich — June 10, 2012 @ 7:07 am

What’s a good literature reference for graph manifolds?

Comment by Mayer A. Landau — June 16, 2012 @ 11:07 am

I don’t know if there is one. You could look at

Behrstock, Jason A.; Neumann, Walter D. Quasi-isometric classification of graph manifold groups. Duke Math. J. 141 (2008), no. 2, 217–240

and the references therein.

Comment by Henry Wilton — June 19, 2012 @ 5:13 am