Some roads are longer than others

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 R3. Fred stated "that sounds like the space of long knots in R3 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 S4 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 S1-family of n-dimensional discs that have a common boundary a "great" (n-1)-sphere. So there is an S1-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 S1-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 S3 in S6 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 R4. 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 S1. 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 Sn+1 deform-spun from knots in Sn? 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 Sn 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 S4 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), S3 is the only lens space that embeds in S4. 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 S4 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 S4.

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 S4. 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 S4 -- 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 R4 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.