curious little things

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:

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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.

A problem I've been interested in recently is to determine the lowest-dimensional Euclidean space a given 3-dimensional manifold embeds into. Hassler Whitney proved all 3-manifolds embed in R^6, and C.T.C. Wall improved that to R^5. Not all 3-manifolds embed in R^4 -- the most elementary obstruction for a closed manifold is orientability, but there are many others. Perhaps the hardest problem in this setting is determining which 3-manifolds embed in R^3.

For example, consider the compact 3-manifolds that have a torus boundary. By Alexander's Torus Theorem, if such a manifold embeds in R^3, it is diffeomorphic to a knot complement in S^3. So how do you determine if a 3-manifold is diffeomorphic to a knot complement?

Technically, there is an answer. Aleksandar Mijatovic says this is how you do it: Triangulate your 3-manifold. Say it has P tetrahedra. Now write a computer program that (a) constructs every knot diagram and (b) produces a triangulation of the knot complement for each diagram. For each knot K, you get a knot complement with B(K) tetrahedra. Now perform all possible Pachner moves on your initial 3-manifold triangulation of length e^{2^{200*P}}*P + e^{2^{200*B(K)}}*B(K) and check if the triangulation of the knot complement appears in this list. If your 3-manifold is a knot complement, this process will terminate.

Although that's an answer, it's not the kind of answer I want. The most superficial criticism is that we do not know how long the computation has to be to ensure your manifold is not a knot complement.

We have a language for 3-manifolds -- geometrization. Ideally we should have an answer couched in that language. Say, ``given an orientable 1-cusped hyperbolic 3-manifold of finite volume, what restrictions on the hyperbolic geometry are equivalent to the 3-manifold being a knot complement?''

A first guess might be that hyperbolic knot complements are precisely 1-cusped hyperbolic manifolds whose fundamental group is generated by conjugate parabolic elements. I do not know how close to the mark this might be.

update (May 13th, 2008): It's totally on the mark. I feel kind of silly for missing the proof, as it's the kind of argument you see over and over again in knot theory textbooks. The easy direction ==> is that every knot complement in S^3 has a presentation of its fundamental group where the generators are meridional elements, which are all conjugate (or conjugate to the inverse of each other). An archetype is the Wirtinger presentation. Combine that with the observation that meridional elements are parabolic in the hyperbolic structure and that completes the argument. For the <== direction -- let M be hyperbolic, 1-cusped with pi_1(M) generated by conjugate parabolics. Do a Dehn filling of M along one such parabolic representative and you have a homotopy 3-sphere. Since the Poincare conjecture is apparently true, the filled manifold is S^3.

Thanks to Ian Agol for pointing this out.

Similar arguments give geometric characterisations of arbitrary link complements in S^3 -- as manifolds whose fundamental group is generated by conjugates of curves on the boundary tori (one for each torus).

To be fair, a criticism of this is that although it's conceptually appealing it's not an algorithm.

Antonio Montalban and Rod Downey have recently proven that, starting with a finitely-presented group, it is generally impossible to compute the 3rd homology group of the corresponding K(pi,1). More precisely, the complexity terminology is that the problem has complexity "Sigma^1_1", roughly translating to "the problem is equivalent to doing a search over a contiumuum for an object that satisfies countably many conditions". The paper can be found on Antonio's webpage, under ``the isomorphism problem for torsion free abelian groups is analytic complete''. In contrast, computing 1 and 2-dimensional homology of a group is easy. 1-dimensional homology is abelianization of the group (whose computation boils down to row-reducing an integer matrix), and 2-dimensional homology is computable via Hopf's formula.

update (July 12th, 2008): I've recently heard that Cameron Gordon has demonstrated that H_3(f.p. group) is `hard' to compute although I have not found a precise reference.

2nd update (September 5th, 2008): This is a surprise. Gordon proved more than I suspected possible. He shows that H_2(f.p. group) is `hard' to compute. Here is the reference: Gordon, C. Some embedding theorems and undecidability questions for groups. Combinatorial and geometric group theory (Edinburgh, 1993), 105--110, London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995.

The corollary of this is that Hopf's formula is not as easy to use as I suspected!

Whether or not the standard of living of North Americans has been decaying or improving or stagnating since the 1970's is a common point of discussion on a lot of blogs. People compare things like wage mobility, median income, starting incomes, home costs...

One point of comparison that has come to seem relevant to me recently is housing cost vs. pay. My father, when he started off in his academic career, was the sole income earner in a family of 4. A new house in the suburbs of Edmonton at that time cost about 2 or 3 times a starting academic's salary. So after taxes, that's about 4 to 6 years of salary. Here in Victoria the cheapest possible house on the market (in Victoria, Oak Bay and East Saanich) costs about 5-6 years of a starting academic's salary. So after taxes, that's 10-12 years of a person's pay.

The easiest point of criticism of this is that Victoria has an inflated real-estate market so this is not a fair comparison. But if I take 3 times my salary and look to see what I could afford in Edmonton, I find a few trailer homes on the outskirts of town. Edmonton's real-estate market is also inflated, as are most markets in western Canada.

Which leads us to the question, is that going to change? I think so. Canada's population is getting older and finding that the cost of living is going up. Everything from food, heating and fuel costs is going up. Since the cost of housing has risen so much relative to wages over the past 30 years, it's pretty common for a person to find that their most valuable asset is their home. So it only makes sense for them to start selling their homes. But people starting off in their careers are finding they can't afford homes at today's prices. So it's a stand-off between the buyer and the seller.

Another point that I hear frequently -- maybe the cost of housing compared to average salaries is not constant. Maybe in 50 years the cheapest home in Victoria will cost $2,000,000 and only professional property managers will be able to afford homes. For example, most people in Europe do not expect to buy a home. They rent, all their lives. Perhaps we are moving towards a model where a select few own, and the rest rent? I also doubt that.

There's a few big differences between European homes and North American homes. The most obvious I noticed immediately on coming to Victoria. Walking up and down the streets one sees a few "heritage" homes, these are frequently some of the oldest buildings in the city, and are relatively well-preserved. But they are made out of wood. And compared to a typical home in Germany, they're garbage.

German homes in comparison are well-insulated, frequently constructed out of solid materials designed to last, such a brick. Canadian homes are essentially disposable items designed with a lifespan of 40 years in mind. A comment one of the unemployed guys on my hockey team made while I was living in Bonn was "perhaps if we made crappier homes we would have more jobs in Germany." The other main ingredient would be the relative density of cities in Europe, and reliance on public transport. Even in countries of relatively high car-ownership rates like Italy and Germany, people tend to live close to the urban core, in high-density homes, which tends to increase the value of urban homes -- this dynamic I don't entirely understand but I suppose transportation costs in Europe might be one of the main factors.

A nice comparison is Edmonton vs. Rome. The metro area of Edmonton is about 9,400 sqkm. Rome has about 5,300 sqkm. But the population of Rome is more than double Edmonton's. A friend named Paolo navigates his car through Roman traffic every day to get to work. I think if I had to do the same for a few years I'd die of a heart attack.

Last night there was a panel discussion at the University of Victoria. The panel consisted of Maher Arar, Monia Mazigh and Stephen Toope. The short story on Arar is that he is a Canadian citizen who was betrayed by his country and shipped off to Syria to be tortured based on false information exchanged between the RCMP and US Immigration and Naturalization. Maher Arar's name has been cleared but the thing that fascinates me is that essentially none of the people responsible have been identified, let alone prosecuted for the role they played in Canada stabbing Arar in the back. Perhaps the Americans involved will never be prosecuted, but at least we as Canadians should get our own house in order.

There is a clear list of leads. The most glaring would be Juliet O'Neill, who reported disinformation on Maher Arar. O'Neill has not revealed her sources to the public. She seems to think it is her duty as a reporter to protect her sources, even if they are simply criminals masquerading as politicians or working for the Canadian security apparatus. A `big but vague' lead would be the RCMP, who fed bogus information to USIN. We really ought to have a list of names: who did what and under who's supervision. We need to determine who was responsible, or else reporters, politicians and security officials might continue to think they will never be held to account regardless of how badly they mess up their job.

Toope was the most informative of the speakers. He made a very interesting point: all the secret evidence about Maher Arar was public knowledge, even back when O'Neill was passing on these accusations that there was evidence linking him to Al Qaeda. The sole purpose of the secret evidence was to serve as cover for people committing these crimes against Arar. Meaning it was largely political: somewhere a politician or security official decided that the perception of Canada participating in the the War on Terror was more important than protecting the rights and freedoms of Canadians. We need to find that person or else we risk creating more.

Arar, Mazigh and Toope focused their attention on the large-scale implications of Canada's terrorism laws and reforming bill C-36. I think that is a fine ambition. But my instincts as a mathematician tell me that if we can not resolve a single case satisfactorily, there is little hope of fixing the whole machine.

I moved to British Columbia in July of 2007, and promptly realized British Columbia is unlike any other place I've lived before. BC is green and the climate is mild. The people are athletic and healthy (at least compared to other parts of Canada), and the culture is mellow and more mediterranean than the rest of Canada. My colleagues and I go out to have a sit-down lunch every day, with tea and coffee afterwards. But I miss the blue skies of Alberta, the thunder and lightning storms, and yes I miss the beautiful frozen winters and the outdoor skating. In my eyes, there's two negatives about BC. The grey, wet winters, and the (seemingly) broken institutions of society. That might sound strange coming from somebody from Alberta, but I'm increasingly convinced it's true -- the institutions of BC are lacking compared to everywhere else I've lived. To me a glaring example is the Insurance Corporation of British Columbia (ICBC).

If you want to drive a vehicle on public roads in BC, you have to have insurance. Provided you're a BC resident, the only company that can legally insure you for your basic insurance is ICBC. ICBC is a provincially-owned government-run monopoly. Insurance rates are set by the BC Utilities Commission.

Consider this. I have an 1140cc motorcycle which has been insured in several places: Rochester NY, Eugene OR, Edmonton AB, Bonn Germany, Paris France and now Victoria, BC. Insurance for this motorcycle was pretty cheap in Alberta at $160/year. In France my bills added up to about 400 EUR per year. In Germany, it only cost 200 EUR per year (both my European policies insured the bike for all Europe and North Africa). My insurance in Eugene and Rochester was a bit more expensive since it was comprehensive insurance. These policies were about USD $700-$900/year. 3rd party liability insurance would have been in the USD $300/year range at that time. In BC minimal insurance for this motorcycle is $1200/year. At first I thought, "this has to be wrong". But I've since learned this is really the insurance rate that is deemed fair in BC. I would like to know why this is considered fair.

To find out, I've been talking with the ICBC Ombudsman, Janet McKennon. She originally tried to convince me that I'm getting higher-quality insurance for my money in BC. I tried to convince her that such a statement was very difficult to confirm, and even if it was true, my BC insurance policy is certainly not worth 8 times what my Alberta policy provided me. For the kind of money ICBC is charging me, I'd expect free fuel, oil, tires, and someone to occasionally come into my apartment and clean up after me. Janet eventually helped me to put in a Freedom of Information and Privacy Act request for the ICBC budget, relating specifically to motorcycles. Janet put me in contact with Deborah Hebert, who responded to my information request. Deborah sent me selected portions of these documents:

ICBC insurance tariff, correction to tariff, rate design volume 1, rate design volume 2.

Page 330 of the rate design volume 2 is one of the most relevant bits. There are three particularly interesting columns. Number of written vehicles is the total number of motorcycle insurance policies. Earned premiums is the amount of money ICBC takes in for the year on its active policies. Claims and adjustment expenses incurred is the amount of money lost on those insurance policies. So if you add up earned premiums - claims and adjustment expenses over the three 2006 rows (third party total, accident benefits, underinsured motorist) you get the net income of ICBC (presumably this is neglecting their other main expenses -- employee salaries and such). But it gives a nice idea of what the ICBC margins are like. So I take my calculator and add things up to get this figure: $374.15 is the net profit ICBC makes per motorcycle policy, per year (based on the 2006 figure). Considering that this number is more than twice what my motorcycle policy cost in Alberta, I would call this amount of profit extremely excessive.

It's curious that if you go back to previous years, you get rather different numbers. Take a look at the claims and adjustment expeses -- from 2002 to 2005 it shows a steady increase, but then in 2006 there is a sudden drop. I have yet to discover the source of this drop. For example, if you compute the per-policy net profit per year in 2005, you get a loss of $12.86, which means motorcycle policies in 2005 were subsidised by other vehicle insurance policies.

To anyone that's interested, the ICBC webpages contains loads of information. This document contains detailed accident statistics for BC are available here. The most interesting aspect of the document is that it shows the number one type of multi-vehicle accident for a motorcyclist in BC is being rear-ended at an intersection. California accident statistics for a long time have the number one multi-vehicle motorcycle accident being the failure of the car to yield the right of way (from the "Hurt Report"). The comparable ICBC stats appear on page 92 of the 2005 report. I think the reason for the difference with the California statistics, is that the ICBC stats separate "Left turn across oncoming traffic" with "intersection - right angle". If you combine those two numbers, you get something that looks more like the Hurt Report.