Yesterday I received the shocking news of the passing of Tim Cochran (1955-2014), a leader in the field of knot and link concordance. The Rice University obituary is here.
A groundbreaking paper which made a deep impression on a lot of people, including me, was Cochran-Orr-Teichner’s Knot concordance, Whitney towers and signatures. This paper revealed an unexpected geometric filtration of the topological knot concordance group, which formed the basis for much of Tim Cochran’s subsequent work with collaborators, and the work of many other people.
In this post, in memory of Tim, I will say a few words about roughly what all of this is about. (more…)
Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”. It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.
I have a rather naive question for the participants here. I’m at the Max Planck 4-manifolds semester, currently sitting through many talks about knot concordance and various filtrations of the knot concordance group.
Do any of you have a feeling for how knot concordance should be organized, say if one was looking for some global structure? In the purely 3-dimensional world there are many very “tidy” ways to organize knots and links. There’s the associated 3-manifold, geometrization. There’s double branched covers and equivariant geometrization, arborescent knots and tangle decompositions. I find these perspectives to be rather rich in insights and frequently they’re computable for reasonable-sized objects.
But knot concordance as a field feels much more like the Vassiliev invariant perspective on knots: graded vector spaces of invariants. Typically these vector spaces are very large and it’s difficult to compute anything beyond the simplest objects.
My initial inclination is that if one is looking for elegant structure in knot concordance, perhaps it would be at the level of concordance categories. But what kind of structure would you be looking for on these objects? I don’t think I’ve seen much in the way of natural operations on slice discs or concordances in general, beyond Morse-theoretic cutting and pasting. Have you?
I’m going to take a break from data topology for this post and write about an interesting construction that I heard Jeremy Van Horn-Morris talk about at the Georgia Topology conference at the beginning of the summer. I should admit that it took me a while to appreciate this definition of a generalized open book decomposition because they only occur in toroidal 3-manifolds with very specific JSJ decompositions. However, they come out of a very natural generalization of 4-dimensional Lefschetz fibrations in which the 3-manifold arises as the boundary of the 4-manifold. These were first developed by Jeremy, Sam Lisi, and Chris Wendl, in a preprint that is still being written. Jeremy and Inanc Baykur  also use this construction to produce contact structures that disprove a number of former conjectures, so even though these 3-manifold are not hyperbolic, they are interesting from the perspective of contact topology. (more…)
One of the foundational papers in Quantum Topology, and one of the main reasons that the subject is called Quantum Topology, is Edward Witten’s landmark paper Quantum field theory and the Jones polynomial. One of the things Witten did in that paper was to define a –manifold invariant as a partition function with action functional proportional to the Chern-Simons –form. A partition function is a path integral, so Witten’s invariant is a physical construction rather than a mathematical one. Quantum topology of –manifolds is, to a large extent, the field whose goal is to mathematically reconstruct, and to understand, Witten’s invariant. Meanwhile, for –manifolds with a metric, Witten defined a –manifold invariant as a partition function in another landmark paper Topological quantum field theory.
I should warn you that I don’t know any physics so some (all?) of what I say below might be rubbish. Still, pressing boldly ahead…
Up until recently, mathematicians only understood tiny corners of Witten’s invariants, or, more broadly, of invariants (topological or otherwise) of manifolds (with or without extra structure) which come from quantum field theory partition functions. But I’ve recently glanced through two papers which seem to finally be going further, seeing more. The tiny corners we have seen already give rise to mathematical invariants of preternatural power (surely that’s the best word to describe it!), such as Ohtsuki series of rational homology –spheres (), Donaldson invariants, and Seiberg–Witten invariants.
Jesse recently recruited me as a special correspondent for the goings-on at Math Overflow. Perhaps he’ll eventually let me blog about other things! To begin I’d like to point out a lovely and easy-to-state but not-so-little problem that appeared on MO.
Is the universal covering of an open subset of Euclidean space diffeomorphic to an open subset of the same Euclidean space?
The above problem is perhaps a representative problem in a family of problems that have received little attention by the geometric topology community, which is the issue of low co-dimension embeddings. They are not well understood. This is because these can be rather difficult problems. More than that, there isn’t an edifice — there’s no standard machine to play with.