homology of knot spaces, known unknowns -- part 1: torsion

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 Rj in Rn 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 E1-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 S1 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 S1 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 S1 (you get them all by turning any long trefoil by 2π about the long axis), so this picture provides an S1 x S1 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 (S1 x S1)/Z2 family of Whitehead doubles of trefoils, and Z2 acts on S1xS1 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 Sj in Sn via a pseudoisotopy sequence, and that's how I found it. The torsion occurs in the homology of the embedding space of Rj in Rn, for j>1 and n-j even. It occurs in H2n-3j-3 which is the first non-trivial homology/homotopy group, and it is Z2. It has a very simple description, too. Take a `long' immersion of R in R3 with two regular double points such that one resolution gives a trefoil, like so:


Now consider this to be an immersion in Rn. The tangent space to the double points are 2-dimensional, so they have an n-2-dimensional complement. This means there is an Sn-3xSn-3-dimensional family of resolutions of this immersion to long embeddings of R1 into Rn. 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 S2n-6 --> long embeddings of R in Rn. Consider a map from S2n-6 to a space X to be a map from R2n-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 R2n-5 into R3n-6, and it is a Haefliger sphere (well, its 1-point compactification, as an embedding of S2n-5 in S3n-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 Rj+1 in Rn+j, and this is also 2-torsion for all j>0 and n odd. Reference.