An embedded circle separates the sphere into two connected components; this is the Jordan curve theorem. A strengthening of this fact, called the Jordan-Schoenflies theorem, says that the two components are disks; i.e. every embedded circle in the sphere bounds a disk on both sides.
One dimension higher, Alexander proved that every smoothly embedded 2-sphere in the 3-sphere bounds a ball on both sides. However the hypothesis of smoothness cannot be removed; in two three-page papers which appeared successively in the same volume of the Proceedings of the National Academy of Science, Alexander proved his theorem, and gave an example of a topological sphere that does not bound a ball on one side (a modified version bounds a ball on neither side). This counterexample is usually called the Alexander Horned Sphere; the `bad’ side is called a crumpled cube. For a picture of Alexander’s sphere, see this post (the `bad’ side is the outside in the figure). The horned sphere is wild; it has a Cantor set of bad points where the sphere does not have a collar; it can’t be smooth at these points.
Let’s denote the horned sphere by and the crumpled cube (i.e. the `bad’ complementary region) by . The interior of is a manifold with perfect infinitely generated fundamental group. itself is not a manifold, but it is simply connected; its `boundary’ is the topological 2-sphere . We can double to produce ; i.e. we glue two copies of together along their common boundary . It is by no means obvious how to analyze the topology of , but Bing famously proved that is . . . homeomorphic to the 3-sphere! I find this profoundly counterintuitive; on the face of it there seems to be no reason to expect is a manifold at all.
There is an obvious involution on which simply switches the two sides; it follows that there is a involution on the 3-sphere whose fixed point set is a wild 2-sphere. Bing’s proof appeared in the Annals of Mathematics; see here. This is an extremely important paper, historically speaking; it introduces for the first time Bing’s `shrinkability criterion’ for certain quotient maps to be approximable by homeomorphisms, and the ideas it introduces are a key part of the proof of the double suspension theorem and the 4-dimensional (topological) Poincare conjecture (more on this in a later post).
The paper is nine pages long, and the heart of the proof is only a couple of pages, and depends on an ingenious inductive construction. However, in Bing’s paper, this construction is indicated only by a series of four hand-drawn figures which in the first place do not obviously satisfy the property Bing claims for them, and in the second place do not obviously suggest how the sequence is to be continued. I spent several hours staring at Bing’s paper without growing any wiser, and decided it was easier to come up with my own construction than to try to puzzle out what Bing must have actually meant. So in the remainder of this blog post I will try to explain Bing’s idea, what his mysterious sequence of figures is supposed to accomplish, and say a few words about how to make this more precise and transparent.
This quarter I’m teaching the “Differential Topology” first-year graduate class, and for a bit of fun, I decided to teach an introduction to characteristic classes, following the classic book of that name by Milnor and Stasheff. The book begins with a discussion of Stiefel-Whitney classes of real bundles, then talks about Euler classes, and then Chern classes of complex bundles, Pontriagin classes, the oriented and unoriented cobordism ring, and so on.
One often-lamented weakness of this otherwise excellent book is that Milnor does not really give much insight into the geometric “meaning” of the characteristic classes; for example, Stiefel-Whitney classes are introduced axiomatically, and then “constructed” by appealing to the axiomatic properties of Steenrod squares, applied to the Thom class. This makes it hard to get a geometric “feel” for these classes, especially in the important case of bundles over a manifold. So I thought it would be useful to give a “geometric” description of Stiefel-Whitney classes in this context (described via Poincaré duality as cycles in the manifold), which is at the same time elementary enough to give a feel, and at the same time is transparently related to the “geometric” definition of Steenrod squares, so that one can see how the two definitions compare.
Because of some turbulence in real life (™) over the last few months, it’s been somewhat hard to concentrate on research. Fortunately the obligation of teaching sometimes exerts the right sort of psychological pressure to keep my mind on mathematics in short bursts. Concerning teaching I believe it was Bott who said (roughly): the trouble with teaching is that when you’ve done it you feel like you’ve accomplished something. But I think this is exactly wrong, and especially absurd coming from a gifted teacher like Bott.
This quarter I’m teaching an introductory graduate class on Kleinian groups. It’s something I could teach standing on my head, and during a couple of the classes I half suspected that I was. But every time I teach something, no matter how “elementary” or “familiar”, I find that I get something new out of it. This time around I have been thinking about the Schläfli formula for the variation of volume in a smooth family of hyperbolic polyhedra, and the way in which it relates to some other well-known and important volume formulae relating to hyperbolic manifolds and geometry, especially in 3 dimensions. It turns out that there are some elegant and easy ways to derive many otherwise quite complicated statements directly from Schläfli; probably this is well known to experts, but it wasn’t to me, and I think it might make an interesting blog post.
Last week at my invitation, David Dumas spoke in the U Chicago geometry seminar and gave a wonderful introductory talk on the theory of convex real projective structures on surfaces. This is the first step on the road to what is colloquially known as “Higher Teichmüller theory”, and the talk made such an impression on me that I felt compelled to summarize it in a blog post, just to organize and clarify the material in my own mind.
Fix a surface (for convenience closed, oriented of genus at least 2). We are interested in the space of convex real projective structures on . This has at least 3 incarnations:
- it is a connected component of the character variety , the space of homomorphisms from into up to conjugacy (note: since all representations in this component are irreducible, one can really take the naive quotient by the conjugation action rather than the usual quotient in the sense of geometric invariant theory);
- it is topologically a cell, homeomorphic to , and can be given explicit coordinates (analogous to Fenchel-Nielsen coordinates for hyperbolic structures) in such a way that a coordinate describes an explicit method to build such a structure from simple pieces by gluing; and
- it has the natural structure of a complex variety; explicitly it is a bundle over the Teichmuller space of whose fiber is isomorphic to the vector space of cubic differentials on .
This last identification is quite remarkable and subtle, since is not a complex Lie group, and its action on the projective plane does not leave invariant a complex structure. Here one should think of as the space of (marked) conformal structures on , rather than as the space of (marked) hyperbolic structures on .
Following Dumas, we explain these three incarnations in turn. Some references for this material are Goldman, Benoist, and Loftin.
The recent passing of Leonard Nimoy prompts me to recall a lesser-known connection between the great man and the theory of (cusped) hyperbolic 3-manifolds, observed by my friend and former mentor Iain Aitchison. In particular, I am moved to give a brief presentation of the (unpublished) work of Aitchison on the theory of manifold-realizable special polyhedral orthocentric curvature-K complexes — or Mr Spock complexes for short. Continue reading
It has been a busy quarter. Since August, I have made 10 trips, to conferences or to give colloquia. On 8 out of the 10 trips, I talked about a recent joint project with Sarah Koch and Alden Walker, on a topic in complex dynamics; our paper is available from the arXiv here. Giving essentially the same talk 8 times (to reasonably large crowds each time) is an interesting experience. The same joke works some times but not others. An explanation that has people nodding their head in one place is met with blank stares in another. A definition passes without comment in one crowd, but leads to a prolonged back-and-forth in another. The nature of the talk (lots of pictures!) meant that I gave a computer talk with slides, so that the overall structure and flow of the talk was quite similar each time; however, I also tried to combine the slides with the occasional use of the blackboard, and some multimedia elements (an animation, an interactive session with a program). I believe my presentation was very similar each time. But my impression of how well the talk went and was received varied tremendously, and I am at a loss to explain exactly why.
In any case, I am officially “retiring” this talk, so for the sake of variety, and while I am still at the point where everything is very coherent and organized in my mind, I will attempt to translate the talk into a blog post.
Posted in Complex analysis, Dynamics, Hyperbolic geometry, Number theory
Tagged Bandt's conjecture, Barnsley-Harrington, complex dynamics, Julia set, landmark point, limit set, Mandelbrot set, Misiurewicz point, pairs of similarities, polynomials, roots, Schottky semigroups, Solomyak, traps
This week I visited Washington University in St. Louis to give a colloquium, and caught up with a couple of my old foliations friends, namely Rachel Roberts and Larry Conlon. Actually, I had caught up with Rachel (to some extent) the weekend before, when we both spoke at the Texas Geometry and Topology Conference in Austin, where Rachel gave a talk about her recent proof (joint with Will Kazez) that every taut foliation on a 3-manifold (other than ) can be approximated by both positive and negative contact structures; it follows that admits a symplectic structure with pseudoconvex boundary, and one deduces nontriviality of various invariants associated to (Seiberg-Witten, Heegaard Floer Homology, etc.). This theorem was known for sufficiently smooth (at least ) foliations by Eliashberg-Thurston, as exposed in their confoliations monograph, and it is one of the cornerstones of -dimensional symplectic geometry; unfortunately (fortunately?) many natural constructions of foliations on 3-manifolds can be done only in the or world. So the theorem of Rachel and Will is a big deal.
If we denote the foliation by which is the kernel of a 1-form and suppose the approximating positive and negative contact structures are given by the kernels of 1-forms where and pointwise, the symplectic form on is given by the formula
for some small , where is any closed 2-form on which is (strictly) positive on (and therefore also positive on the kernel of if the contact structures approximate the foliation sufficiently closely). The existence of such a closed 2-form is one of the well-known characterizations of tautness for foliations of 3-manifolds. I know several proofs, and at one point considered myself an expert in the theory of taut foliations. But when Cliff Taubes happened to ask me a few months ago which cohomology classes in are represented by such forms , and in particular whether the Euler class of could be represented by such a form, I was embarrassed to discover that I had never considered the question before.
The answer is actually well-known and quite easy to state, and is one of the applications of Sullivan’s theory of foliation cycles. One can also give a more hands-on topological proof which is special to codimension 1 foliations of 3-manifolds. Since the theory of taut foliations of 3-manifolds is a somewhat lost art, I thought it would be worthwhile to write a blog post giving the answer, and explaining the proofs.