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?

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The smooth and topological theories are different, if related animals. Here’s a very rough idea of the picture that has emerged

Topological concordance is a 4-dimensional surgery problem. One let’s surgery on the knot bound a 4-manifold, and tries to measure the obstruction to changing this to a homology circle. Roughly, one expects these obstructions will lie in 1-1 correspondence to concordance classes, as they do in high dimensions.

One would like to define “homology surgery obstruction groups” for 4-manifolds, which simplifies in high dimensions to the Cappell-Shaneson surgery theory used to classify homology cobordism of high dimensional manifolds. But things seem much more complicated. In particular, in dimension 4, Freedman tells us we can iterate Whitney disks, and again roughly speaking, if we do so with sufficient control we can find embedded disks in a compactified regular neighborhood of the tower of Whitney disks. The Whitney trick allows us to embed spheres in homology classes, and excise 2-dimensional homology classes through surgery.

The filtrations arise from this perspective. One builds a Whitney tower one stage at a time, sometimes unbuilding and rebuilding to make the tower higher. This recursive process is reflected through the filtrations on the concordance group.

The tricky part arises because the group of the 4-manifold and the three manifold boundary have closely related fundamental groups, a connection captured through various versions of Poincare duality (linking and intersection theory.) This is strikingly different than in high dimensions, where all slice knots are slice with complements that have fundamental group the integers. Understanding this connection deeply lies at the heart of building and computing these filtrations. This connection of the fundamental groups of the 3 and 4 manifolds began with Casson and Gordon, and their seminal concordance invariants. Work of others since has elaborated that model, a non-trivial task.

One might hope to come from the other end, and build a surgery group globally instead of recursively, that is, instead of building approximations of surgery groups via invariants indexed on filtrations. Mark Powell and I have strong ideas of how to do this, and hope to publish something eventually. But this, like the filtrations approach, still must detect the way the fundamental group can change under concordance, and the resulting theory will be quite abstract. Only time will tell how computable it may be.

As for smooth concordance, as in high dimensions, one ideally wishes to build a “smoothing theory” which determines concordance classes of knots in a given topological concordance class. By additivity, it would suffice to compute smooth concordance of topologically slice knots. But modern 4-dimensional smoothing theory has resisted this approach, and one can’t know if this sort of 4-dimensional smoothing theory exists. We’re still trying to understand any good global picture of smoothing theory.

The Cochran-Harvey-Horn bi-polar filtration is an admirable step in the direction of conceptually merging topological and smooth concordance theory.

Comment by Kent Orr — May 16, 2013 @ 12:54 pm |

Thanks for giving me your perspective Kent. Where does the Cappell-Shaneson result appear, regarding knot concordance in high dimensions? I’m paging through the mathscinet Cappell-Shaneson papers with no luck so far.

Comment by Ryan Budney — May 17, 2013 @ 9:32 am |

“The Codimension Placement Problem and Homology Equivalent Manifolds.” Ann. of Math. (2) 99 (1974), 277–348. It’s the beginning of a long string of papers on embedding and immersion theory. Their paper “Link cobordism” may be a little easier to read.

Comment by Kent Orr — May 18, 2013 @ 8:49 am

Let me answer Ryan’s question with a shameless bit of self-advertisement.

Peter Teichner and I wrote a while ago a paper with the title `New topologically slice knots’.

In this paper we propose the following conjecture for when a knot is topologically slice.

Let K be a knot in S^3. We denote its zero-framed surgery by N_K.

Then we conjecture that K is topologically slice if and only if there exists a an epimorphism from pi_1(N_K) onto a ribbon group,

(i.e. a group with a Wirtinger presentation of deficiency one and with abelianization Z) such that

Ext_Z[G]^1(H_1(N_K,Z[G]),Z[G])=0.

The condition looks funny, but for G=Z this is just the condition that the Alexander polynomial vanishes.

The `if’ direction of the conjecture was shown for G=Z by Freedman and for G=Z \ltimes Z[1/2]

in our paper. It is difficult to make much progress on the conjecture as long as we don’t know whether surgery works for all groups.

The `if’ direction has been shown precisely for the two ribbon groups which are solvable, i.e. for which surgery is known to work.

Comment by Stefan Friedl — May 16, 2013 @ 10:25 pm |