curious little things

My take, written up here.

If clothing manufacturers made pants like these, well, you wouldn't be able to wear them inside-out.

Towards the end of June the University of Brisbane is hosting a conference on a subject which as the quote above suggests, is something of a barren landscape scattered with unsavory characters.

In my mind the motivation for this conference comes from two sources, one is entirely structuralist, and another comes from a remarkable theorem. The remarkable theorem is Rubinstein's 3-sphere recognition algorithm. Let me state it:

Theorem: There is an algorithm to determine if a compact triangulated 3-manifold is homeomorphic to the 3-sphere.

The really remarkable aspect of this theorem is the statement of the algorithm. The algorithm (as currently implemented in Ben Burton's software Regina) has an exponential run-time in the number of simplices in the initial triangulation. It can also be fairly memory intensive.

Algorithm: Start with your initial triangulation and enumerate all normal 2-spheres. Crush the normal 2-spheres until you have no more normal 2-spheres available to crush. This converts your original 3-manifold into a wedge of 3-manifolds. For each one of these wedge summands, check to see if there is an almost-normal 2-sphere. If one of the wedge summands fails to have an almost-normal 2-sphere, your original manifold (and that wedge summand) is not the 3-sphere. If every wedge summand has an almost-normal 2-sphere, they're all 3-spheres and your original manifold is the 3-sphere.

If the terminology normal and almost-normal is unfamiliar, the basic idea is that normal surfaces in a triangulated manifold are the ones that "look linear" inside of a simplex. So enumeration of these turns into an integer linear-programming problem. Almost-normal surfaces are an invention of Rubinstein's -- the idea is that normal surfaces are nice, but if you want to know isotopy relations, or embedded surgery relations between surfaces, they do not suffice. Almost normal surfaces are the "critical" surfaces in isotopies or embedded surguries between normal surfaces. There is a direct analogy from "Morse Theory to Cerf Theory" as with "Normal Surface Theory to Almost Normal Surface Theory."

So the idea is this: if 3-sphere recognition is so beautiful and algorithmic, shouldn't there be some aspect of this technology that should work for 4-manifolds? We want to know.

And that gets to the structuralist aspect of this conference. Triangulations of 4-manifolds have not been used much at all in the field. So they introduce a new set of biases and prejudices, rather different from, say, the surgery approach. That's enough for me.

Conference webpage is here.

IPMU (the Institute for the Physics and Mathematics of the Universe) officially opened its new building this week. Ben Burton and I were in Tokyo visiting and I took some photos.

The building is quite nice -- spatious, comfortable offices, many small seminar rooms. And tea time every day at 3pm. Unfortunately we left Tokyo two days before the official opening but it was a productive week for us.

These three photos are looking out from my office into the main atrium, left-to-right. It gives a rough sense of the building.

Now *that's* a creation with real merit!

Maybe a better title would have been "why I'm not a cartoonist."

Back when I was a grad student, the Lawrence-Krammer representations of the braid groups were a relatively new thing. These representations were apparently known to Magnus, although the first person to really study them was Ruth Lawrence, who demonstrated how to compute the Jones polynomial of a knot from a plat closure using the Lawrence representations. Daan Krammer rediscovered the Lawrence representations in his search for a faithful representation of the braid groups, then he and Bigelow proved they were faithful -- by rather beautifully independent methods. Back when I was a grad student I was hopeful that I could use the Lawrence-Krammer representation to construct some new invariants of knots or 3-manifolds. I had observed that the Lawrence-Krammer representation preserved a sesquilinear form, and I was hoping to use that form to "mine" the representation for invariants.

Some specifics. The Lawrence-Krammer representation is the representation from the Braid group on n strands to the general linear group of rank "n choose 2" matrices with entries in the group ring Z[ZxZ]=Z[q,t] -- ie: 2-variable Laurent polynomials with integer coefficients. It comes about from the braid group's action on the Lawrence-Krammer module, which is the 2nd homology group of a ZxZ cover of the configuration space of 2 points in a punctured disc (as many punctures as there are strands in the braid). As a module over the covering transformations, the homology has rank n choose 2, and we label the generators by v_{j,k} where 1 <= j < k <= n. The matrices have the form:

The sigma_i's are the Artin generators of the braid group. The form preserved by the Lawrence-Krammer representation is given by:

One of the striking things about this sesquilinear form (and the representation) is that when the variables "q,t" are specialized unit complex numbers, the representation becomes a complex representation, and the sesquilinear form becomes a non-degenerate Hermitian form. The signature of the form is not constant and it has a "shattered glass" appearance:

The above image is a sketch of the signature vs q and t in the case of the 6-stranded braid group. So when q is close to 1 and t close to i, the representation is negative-definite and one can think of the braid group as a subgroup of a unitary group. This is reminiscent of how the free group on 2 generators is a dense subgroup of SO_3 -- as in the proof of the Banach-Tarski paradox.

Anyhow, why was I studying this again? Matt Zinno had shown that the Lawrence-Krammer representation is irreducible. I was hoping that if one restricted the representation to suitable natural subgroups -- like say the Hilden or "Wicket" subgroup as some people like to say, then the representation would reduce and one might be able to derive knot and link invariants from this, by means of the Birman-Hilden correspondence between links and their plat closures. Alas, the Lawrence-Krammer representation turns out to be irreducible over the Hilden/Wicket subgroup as well.

This result has been rediscovered recently by Stoimenow. Stoimenow has gone much further to show that the image of these representations of the braid group is dense in the unitary groups. So among other things, the above sesquilinear form preserved by the Lawrence-Krammer representation is essentially the "only" one that can be preserved.

Now I'm interested in a different analogy. The representations of the mapping class groups of a surface on the homology of the surface preserve the intersection product (the dual of the cup product via Poincare duality). Moreover, the image of these representations into the automorphisms of the homology is precisely the intersection-product preserving automorphisms. So perhaps something like this theorem is true for the braid group. Ie consider the Lawrence-Krammer representation to be a homomorphism:

B_n --> GL_{Z[q,t]} (n(n-1)/2)

Let L_n be the subgroup of GL_{Z[q,t]} (n(n-1)/2) which preserves the above intersection form. So B_n --> L_n is injective. How `far' is this map from being surjective?

L_n is strictly larger than B_n since L_n contains the full centre of the general linear group, yet the image of B_n only contains a `slice' of it, as Bigelow observed in his dissertation. Is there much more to this story?

Fois gras is sort of like a high-end Vegemite.

Like Vegemite, fois gras is not appreciated by many people. It can be strong tasting, and people tend to be put-off when they hear how either product is made.

While fois gras is fatty, Vegemite is salty. While fois gras is considered by some to be a delicacy, Vegemite is, uh, not. And while fois gras is sort of like eating fat, Vegemite is sort of like eating bread or drinking beer.

This post is dedicated to Pascal Lambrechts.

The creative process for papers has always been murky to me. I rarely set out to write a particular type of paper. Quite often results appear as by-products of computations I'm working on for some other reason.

I've been trying to make a paper readable. Meaning, I've been writing it for quite a while and it's more or less done, but I need to make it presentable. It's on embeddings of 3-manifold in the 4-sphere and it has had the longest incubation-period of anything I've ever written. I've wanted to work on this topic for quite some time but it was only when I moved to the Max Planck Institute that I started to take the topic seriously. So I've been working on this paper on-and-off for the past 3 years now.

My first paper was with Stephen Bigelow. I had been studying a problem called the "generalized Smale conjecture" for spherical 3-manifolds, largely from the perspective of Bonahon, Rubinstein and Scharlemann's work on Heegaard splittings, sweep-outs, etc. In the process of studying Heegaard splittings I had become acquainted with Birman's work on mapping class groups, branched covering spaces, normalizers and so on. When Bigelow gave his talk at the Cornell Topology Festival on linearity of the braid groups, it was natural to talk with him about extending the result to other mapping class groups. We went on a walk at Tremen Park and the paper was born. It was pretty much that simple -- although during the writing-up process Stephen noticed that we could cut the dimension of the representation down to 64 if we were a bit more careful with our choice of group extensions.

In contrast, "Little cubes and long knots" was entirely unanticipated. The germinal moment for the paper came from conversations I had with Fred Cohen while a postdoc at Rochester. Fred and I would regularly chat about mathematics in his office and we'd play around with a variety of topics. I was telling him about some observations Hatcher made, that braid groups appear in the fundamental group of certain components of the space of long knots in R^3. Fred stated "that sounds like the space of long knots in R^3 has an action of the operad of 2-cubes", and he thought maybe Victor Turchin had proven such a statement. At the time I knew little at all about operads. Fred asked me if I could construct such an action. In the ensuing weeks I tried to massage the long knot space into something where the operad of cubes acted. In the process I would show Fred various candidates and he would point-out my mistakes. I also wrote Victor and found out that he conjectured that 2-cubes (or some equivalent operad) acted on the space of long knots but he did not have a proof. It was about that time that Fred stopped finding errors in my most recent constructions and I started to feel confidant I had an action. The action turned out to be more general than we had anticipated in that it engulfed what's known as the "Cerf-Morlet comparison theorem" in that it showed many embedding spaces and diffeomorphism groups have actions of operads of cubes and are frequently iterated loop-spaces. This gave me a lot to think about because the Cerf-Morlet comparison theorem has always been mysterious to me. Things evolved from there. It seemed the 2-cubes action "said alot" about the space of long knots in R^3 and I wanted to quantify that somehow. So I asked Fred if there was a notion of "free 2-cubes object" and it turns out there is. So then we started to play around with the idea that maybe the long knot space is a free 2-cubes object. We invited Hatcher up to visit and we kicked around the question a bit. We came to the conclusion that it looked "reasonable". When school ended I went up to Ottawa to spend the summer with my sister. That gave me a pretty relaxed atmosphere to pursue this question. During the day I helped Richard (Jen's boyfriend at the time) to destroy/rebuild their house or I'd walk their puppy. Then in the afternoon I thought about proving freeness. Mid-way through the summer I "saw" the proof in a flash. There were a bunch of details that had to be filled in -- most of them centering around my then foggy understanding of exactly what the JSJ-decomposition did for knot complements. But that would happen over the next few months as I wrote down the details of the proof.

My most recent paper on embeddings of 3-manifolds in the 4-sphere is something completely different. First, there's no major theorems. As a paper, it's a massive pile of small observations ranging from standard applications of standard tools, to some slightly novel applications and a scattering of some slightly novel constructions. The goal of the paper is not so much the resolution of the embeddings problem (because I don't know how) but more to simply create a list -- something that allows us to measure our progress on the problem. Much of my motivation is that I had little in the way of context for judging how difficult this topic of embedding 3-manifolds in S^4 is.

It's an interesting process how we come to decide if a topic is worthy of study. For me, it came about as a natural progression from studying spaces of knots. There is a beautiful and underexploited connection between "spaces of knots" issues and plain old classical knot theory, called spinning. Originally spinning was due to Artin -- his process took as input a co-dimension 2 knot in the n-sphere and produced a new co-dimension 2 knot in the (n+1)-sphere whose complement has the same fundamental group as the "old" knot complement. Artin's spinning goes like this: think of the (n+1)-sphere as being swept-out by an S^1-family of n-dimensional discs that have a common boundary a "great" (n-1)-sphere. So there is an S^1-family of isometries of the (n+1)-sphere given by rotation about this great (n-1)-sphere. Put a "long" co-dimension 2 knot in the n-disc -- ie, make sure its boundary is a (n-3)-sphere in the "great" (n-1)-sphere. Then apply the S^1-family of rotations. This sweeps-out a co-dimension 2 knot in the (n+1)-sphere. Zeeman expanded this notion of spinning to "twist-spinning" by using a less rigid sweep-out process. During the sweep-out, Zeeman allowed rotation about the "long axis" represented by the standard (n-2)-disc in the n-disc. Litherland went one step further and allowed any motion of the knot being "graphed" in this sweep-out process. In a broad sense, Litherland's version of spinning should probably be thought of as a hybrid of Alexander's theorem that links in the 3-sphere have a closed braid form, and Artin's spinning construction.

I had been studying spaces of knots for several years and was surprised that I hadn't heard much in the way of significant results on this spinning process. As far as I knew, it was used to construct knots but there seemed to be a "theorem deficit" on the topic. A result in "a family of embedding spaces" caused me to take the construction more seriously. The Litherland spinning construction makes sense for more than co-dimension 2 knots, and I showed that provided the co-dimension is greater than 2, all knots are deform-spun. Not only that, knots are frequently "multiply deform-spun". In a previous post I went into some detail on this -- an archetypal example is that the embeddings of S^3 in S^6 are "double spun" in the sense that they are obtained by graphing elements from the 2nd homotopy group of the space of long embeddings of R into R^4. This is all a reflection on the larger fact that the deform-spinning process "is" the boundary map in the pseudo-isotopy fibration for embedding spaces. Moreover, my proof was totally elementary. If you look back at Haefliger's first paper on high co-dimension knot theory, he shows that the isotopy classes of knots form a group under the connect-sum operation. My proof is simply his proof, but I force his concordance argument into the context of pseudo-isotopy embedding spaces. This restructuring of his argument gave it the extra geometric strength.

So I became convinced that deform-spinning is elementary and worthy of serious study. I started poking at the topic. I learned about Litherland and Zeeman's work only after proving the above theorems on deform-spinning. It was about this time that I realized how powerful MathSciNet is for not only paging backwards through the history of a topic, but to find the papers written after a given paper that make reference to it. Thanks MathSciNet! Litherland and Zeeman's main result is that co-dimension 2 deform-spun knots frequently have complements that fibre over S^1. Litherland described the Seifert surface for the deform-spun knot in a way that's sort of a combination of an open-book decomposition and a cyclic branch covering space construction. I'll come back to this in a few paragraphs.

Since deform-spinning is so "rich in inputs" it led me to a rather simple question: are all co-dimension 2 knots in S^(n+1) deform-spun from knots in S^n? This question recalls the work of Kervaire, Yajima, Fox and Levine on the fundamental groups of complements of knots. The upshot of their work is that this class of groups increases as n increases. But once one is considering co-dimension 2 knots in S^n for n>=5, it stabilizes on a well-known class of groups. For n<5 the main tool used to distinguish such classed of groups is the Alexander module -- and specifically the aspects of this module most closely connected to Poincare duality. This led to my paper with Mozgova where we showed that not all knots in S^4 are deform-spun. I think there are likely to be many other obstructions for knots to be deform-spun but as the dimension increases likely these will be more difficult to find. I'm a little hopeful that when n is large enough, all knots will be deform-spun, or at least the class of deform-spun knots may be easily recognisable. In a rough philosophical sense, such a result would be a knot-theoretic analogue to Cerf's pseudoisotopy theorem.

edit: (Aug 17th, 2008) There's *lots* of obstructions to higher-dimensional knots being deform-spun. The ones I observed today come from Poincare duality on the Alexander modules of the knot. PD leads to some strong divisibility conditions on the Alexander polynomials and further symmetry conditions, generalizing the paper with Mozgova.

I got interested in studying embeddings of 3-manifold in the 4-sphere via this simple contrast: by a Poincare Duality argument (originally due to Hantsche), S^3 is the only lens space that embeds in S^4. But it was observed as early as Zeeman that the connect sum L#-L of one lens space with its orientation-reverse smoothly embeds in S^4 provided the order of the fundamental group of L is odd. Moreover, Fintushel and Stern went on to prove the converse. The embedding is readily visualizable as the Zeeman-Litherland Seifert surface for deform-spun knots obtained by 2-twist spinning 2-bridge knots. Fintushel and Stern's result is the culmination of some pretty powerful invariants -- on top of Zeeman's pretty powerful technique for constructing embeddings of 3-manifolds in S^4.

I think the above example is probably indicitive of how the story will play out if it's pursued further. The Zeeman-Litherland Seifert surface construction is pretty specialized. I doubt there will be one simple and uniform technique for constructing embeddings of all 3-manifolds in S^4. Likely techniques will have to be heavily adapted to pretty specific and specialized classes of 3-manifolds. Of course, there could very well be a nice constructive technique to find embeddings of all 3-manifolds that embed in S^4 -- but likely the technique would only be useful for individual 3-manifolds, not for theoretical results about families of 3-manifolds. An avenue that would be interesting to explore might be generic maps of 3-manifolds into the complex plane C. ie: consider an embedding of a 3-manifold M in R^4 to be a special pair of generic maps M --> C. There is a fairly elaborate description of generic/stable maps from 3-manifolds into the plane due to people like Boardman, Thom, H.Levine and others. This technology has been used recently by D. Thurston and F. Costantino to give effective procedures for 3-manifold cobordism problems. So it seems like the stage is set.

Some nice news -- Antoni Kosinski's Differential Manifolds book has been Dovered. The review on Amazon is pretty honest in that it's a book that's not without its problems. I don't know if it's me, but that just makes the book more readable. I've always had trouble reading Milnor's books because there aren't enough errors to keep me entertained. I like books to present the big idea, give me a few clues and then let me work out the details.

Kosinski's book has some typos and some weirdnesses. One weirdness that I found a bit `above and beyond' is that the book has a long and careful build-up to the h-cobordism theorem, but when it gets to the big magic Whitney trick moment, it defers to Milnor's h-cobordism notes... Milnor's notes for a long time had been out of print... which really annoyed me at the time. Luckily, those notes are readily found nowadays by anyone who can use a Google prompt. Alternatively, the Whitney trick can be learned from Whitney's papers.

Perhaps the thing that Kosinski's book does best is show how one can systematically avoid the dreaded `straightening the corners' problem that is inherent in Smale's h-cobordism proof. For that, it's worth a read.

There's a fair amount of papers in the literature on the homotopy and homology of spaces of knots. An oddity that probably isn't apparent to the casual reader is that very little is known about torsion in the homology of knot spaces. To be precise, let's `normalize' this discussion and consider knot spaces to be the space smooth of embeddings of R^j in R^n which agree with a fixed linear embedding outside of a fixed ball. When j=1 and n>3, it is not known if that space has any torsion elements in its homology. Much is known about the homology spectral sequence -- it converges, among other things. Pascal Lambrechts, Victor Turchin and Ismar Volic have recently shown that the rational spectral sequence collapses at the E^1-term. Other than the fact that no torsion has been demonstrated, many other mysteries remain -- ie: even though the rational spectral sequence collapses, we still do not `know' the homology of these embedding spaces, since all we have is a DGA whose homology agrees with the homology of the knot space. It is still potentially `a lot of work' to compute the homology of this DGA in any meaningful way.

Fred Cohen and I have demonstrated that for j=1 and n=3, there's all kinds of torsion in the homology of the knot space. The easiest way to see it is to consider the component of the long knot space corresponding to the Whithead double of a trefoil knot. It turns out this component has the homotopy-type of S^1 x klein bottle. The Klein bottle has 2-torsion in its first homology group. The way to think of that 2-torsion is to consider the Klein bottle to be fibred over S^1 with fibre a circle. The monodromy flips the fibre. Now to see this in the long knot space, consider the patterns that generate all Whitehead doubles of the trefoil. Here is a 1-parameter family of `long' Whitehead links.

.

There is the long component, and the closed component. The closed component is a round embedded circle, which bounds a flat disc. Cut the long component by the flat disc, grab the bloody ends and tie a trefoil into them when re-gluing them together. The space of long embeddings of a trefoil knot turns out to have the homotopy-type of S^1 (you get them all by turning any long trefoil by 2\pi about the long axis), so this picture provides an S^1 x S^1 family of Whitehead doubles. The trefoil is strongly invertible, so put it into such a position -- then by the symmetry of the above diagram, we get an (S^1 x S^1)/Z_2 family of Whitehead doubles of trefoils, and Z_2 acts on S^1xS^1 as the orientation-covering transformations of the Klein bottle. That's the most accessible torsion in the homology of knot spaces. Reference.

I've also found some torsion in the homology of knot spaces for j and n with j>1. This torsion turns out to be directly related to Haefliger's torsion isotopy classes of embeddings of S^j in S^n via a pseudoisotopy sequence, and that's how I found it. The torsion occurs in the homology of the embedding space of R^j in R^n, for j>1 and n-j even. It occurs in H_{2n-3j-3} which is the first non-trivial homology/homotopy group, and it is Z_2. It has a very simple description, too. Take a `long' immersion of R in R^3 with two regular double points such that one resolution gives a trefoil, like so:

.

Now consider this to be an immersion in R^n. The tangent space to the double points are 2-dimensional, so they have an n-2-dimensional complement. This means there is an S^{n-3}xS^{n-3}-dimensional family of resolutions of this immersion to long embeddings of R^1 into R^n. It turns out this class generates the homology of this embedding space in dimension 2n-6. Also, this is the first non-trivial homology/homotopy group, so one can collapse the 2n-7-skeleton and convert this class into a map S^{2n-6} --> long embeddings of R in R^n. Consider a map from S^{2n-6} to a space X to be a map from R^{2n-6} to X which is constant the base-point outside of some fixed ball. Thus if we `graph' this map, we get an embedding of R^{2n-5} into R^{3n-6}, and it is a Haefliger sphere (well, its 1-point compactification, as an embedding of S^{2n-5} in S^{3n-6} is Haefliger's sphere). When n-j is even, this is a torsion class.

But we can get far more mileage out of this construction. Rather than graphing the whole family, we can do a Fubini-type construction and only graph it part-way. This gives elements in the 2n-j-6-th homotopy group of the space of long embeddings of R^{j+1} in R^{n+j}, and this is also 2-torsion for all j>0 and n odd. Reference.

With the proof of the Poincare conjecture, it's a great time to look at the whole of 3-manifold theory and poke at it. Some things are "more proven" than others. For example: We only have one proof of the Poincare conjecture. Similarly, there's only one proof of the Smale conjecture. Some things are "super proven", like Dehn's Lemma, the Loop Theorem and the Sphere Theorem -- there's even more than one proof of their equivariant versions.

Take a look at the classification of Seifert fibred manifolds. The classification of the "sufficiently large" ones is a rather elegant demonstration of incompressible surfaces. But the small Seifert fibred manifolds -- manifolds that fibre over S^3 with 3 or less singular fibers, their classification is rather fussy and involves a bunch of special cases.

At the moment my favorite proof of the classification of lens spaces is due to:

Przytycki, Yasuhara. Symmetry of Links and Classification of Lens Spaces. Geom. Ded. Vol 98. No. 1. (2003)

Actually, there's aspects of their proof I'd change if I was presenting it myself. But the general idea is what I like. Take a knot K in a lens space L_{p,q} such that its lift to S^3 is a knot with a trivial Alexander polynomial. Now consider K to be a generator of H_1(L_{p,q}). Compute the torsion linking form on (K,K). This is an element of Q/Z. The classification boils down to computing this number and checking that it has little dependance on the choice of K.

The first proof of the classification that I read (and liked) is due to Francis Bonahon.

F. Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), 305--314.

His technique is to show that lens spaces have a single genus 1 Heegaard splitting up to isotopy. He uses a variant of a standard incompressible surface argument, but adapted to the slightly more awkward situation where the surface is the 2-skeleton of the "standard" CW-decomposition of the lens space.

Paolo Salvatore and Riccardo Longoni had a beautiful related insight into lens spaces recently. They showed that even though the lens spaces L_7,1 and L_7,2 are homotopy-equivalent, their configuration spaces are not homotopy-equivalent. Moreover, they only need the homotopy type of C_2(L_{p,q}) -- the configuration space of two points in the lens space. This could potentially give a new and rather elegant classification of the lens spaces if this were true:

Two lens spaces L_{p,q} and L_{a,b} are diffeomorphic if and only if C_2(L_{p,q}) and C_2(L_{a,b}) are homotopy-equivalent.

Shortly after Paolo and Riccardo put their paper on the arXiv I ran into several different groups who said they were thinking about extending the Salvatore-Longoni result, but I haven't heard much positive or negative from any of them since, and it's been almost 4 years now.

I do not know of much work towards simplifying the classification of Seifert-fibered manifolds that fibre over S^2 with 3 singular fibres.