Different flavours of Vassiliev Conjecture

Question

There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of results around Vassiliev invariants that I am not sure how to connect.

Genuine Vassiliev's Conjecture [GVC]

Let $\mathcal{K}$ be the space of knots in 3 dimensions. Let $\sim_n$ be the equivalence relation on $\pi_0(\mathcal{K})$ that identifies knots that cannot be distinguished by Vassiliev invariants of order $\le n$. We have a tower of surjections: $$ \pi_0(\mathcal{K}) \to \ldots \to \pi_0(\mathcal{K})/ \sim_n \to \pi_0( \mathcal{K})/\sim_{n-1} \to \ldots \to \pi_0(\mathcal{K})/ \sim_1 $$ The Genuine Vassiliev Conjecture states that two different knots can always be distinguished by a Vassiliev invariant. In other words, we require the injectivity of the maps $$ \pi_0(\mathcal{K}) \to \varprojlim_{n \to \infty} \pi_0(\mathcal{K})/\sim_n $$

Chord Vassiliev's Conjecture [CVC]

A first line of results in this direction are given by things like Konsevitch Theorem. If $\mathcal{V}_n$ is the set of Vassiliev invariants of order $\le n$, Konsevitch proved that

$$\mathcal{V}_n \otimes \mathbb{R} \cong \bigoplus\limits_{s \le n} A^*_s \otimes \mathbb{R} $$

where $A_s$ is a suitable algebra of chord diagrams. There is a little trick we can do to reformulate Konsevitch theorem in terms of our tower. If we consider $\mathcal{V}_n \otimes \mathbb{Q}$ as a subspace of the dual of $\pi_0(\mathcal{K}) \otimes \mathbb{Q}$, we can write (see here): $$ (\pi_0(\mathcal{K})/ \sim_n ) \otimes \mathbb{Q} \cong \frac{\pi_0(\mathcal{K}) \otimes \mathbb{Q}}{(\mathcal{V}_n \otimes \mathbb{Q} )^{\perp}} \cong (\mathcal{V}_n \otimes \mathbb{Q} )^* \cong \bigoplus_{s \le n} A_s \otimes \mathbb{Q} $$ De Brito and Horel state that there exists a map $A_n \to \ker( \pi_0(\mathcal{K})/\sim_n \to \pi_0(\mathcal{K})/\sim_{n-1} )$, where knots up to $n$-equivalence can be regarded as an abelian group under connected sum. When tensoring over $\mathbb{Q}$, it becomes an isomorphism; this fact, by an easy induction, yields the above formula for $\pi_0(\mathcal{K})/\sim_n$. Thus we will call the Chord Vassiliev's Conjecture the fact that $A_n \to \ker( \pi_0(\mathcal{K})/\sim_n \to \pi_0(\mathcal{K})/\sim_{n-1} )$ is an isomorphism.

Tower Vassiliev's Conjecture [TVC]

At last, there are a few results linking the above tower with the taylor tower of knots. Specifically, Budney et al. proved that there exist maps $\pi_0(\mathcal{K})/\sim_n \to \pi_0(T_{n+1} \mathcal{K})$ that commutes with the tower maps. We call Tower Vassiliev's Conjecture (TVC) the fact that such a map is an isomorphism.

Implications

Let us denote by [CHL], [CCH], [CHT] the collapse at the second page of the homology, cohomology and homotopy spectral sequence associated with the taylor tower, at least over the antidiagonal.

I want to prove some implications between the three vassiliev conjectures, eventually assuming some of the collapsing conditions. Here is what I have tried:

• [TVC] implies [GVC]. Since we have in isomorphism of towers, we can analogously show the injectivity of the map $$\pi_0(\mathcal{K}) \to \varprojlim_{n \to \infty} \pi_0(T_n \mathcal{K} )$$, which is the composite of $\pi_0(\mathcal{K}) \to \pi_0(T_{\infty} \mathcal{K})$ with the projection $$\pi_0(T_{\infty} \mathcal{K}) \to \varprojlim_{n \to \infty} \pi_0(T_n \mathcal{K})$$ I suspect that these maps should be injective for a general reason, which I can't see. For example, it should be the case that $\pi_0(T_{\infty} \mathcal{K})$ is uncountable and $\pi_0(\mathcal{K})$ is countable, which makes injectivity plausible. Unluckily, it seems to me that the connectivity estimates of Goodwillie are not enough to conclude its injectivity. The injectivity of the other map should have to do with the relation between the total homotopy spectral sequence and the truncated ones (converging to $T_n \mathcal{K}$). It seems to me that both sides are given by $ \oplus_p E^{-p,p}_{\infty}$ where the latter is the homotopy spectral sequence; but $\pi_0$ of a homotopy limit is not always the limit of $\pi_0$, so I am definitely missing something.
• [TVC] implies [CVC]. The map reformulates as $ A_n \to \ker( \pi_0 T_{n+1} \mathcal{K} \to \pi_0(T_n\mathcal{K}))$. We know that in the homotopy spectral sequence $ A_n \cong E_{n+1,n+1}^2$ and also that $E^1_{n+1,n+1} = \pi_0( \textrm{fib}(T_{n+1}\mathcal{K} \to T_n \mathcal{K} ) )$. I suspect that using the long exact sequence for the fiber seq and analyzing a bit the spectral sequence should give the result, maybe up to assuming something else on the way.
• [TVC] implies [CHT]. This is stated explicitly by De Brito-Horel in theorem 6.1.
• [CCH] implies [CVC]. Indeed, Vassiliev invariants are $\oplus_{p} E^{-p,p}_{\infty}$, where the latter is the cohomology spectral sequence, while chord algebras pop up at the second page of Sinha Spectral sequence (first of Vassiliev) on the same line.

Questions

1. How many wrong things have I said? Can you fill in the gaps in my implications?
2. Are there other implications that I am missing?
3. Could you share your perspective on this matter? I feel like mine is somewhat goofy deformation of a solid one.
4. How does the homology spectral sequence fit in this framework? Why has it been so intensively studied?

Thank you. Of course, since it is a lot of material, partial answers are ok too.

Answer 1

Here are some coments :

1. [TVC] implies [GVC]. I don't think this is easy. In fact Danica Kosanovic has proved, in her thesis, the surjectivity part of [TVC]. So proving that knots can be distinguished in some stage of the Goodwillie-Weiss spectral sequence could be harder than [GVC]. I don't think it's known that either of the maps that you mention are injective, although, the second one has a chance of being injective for general reasons (you would have to analyse the $lim^1$-term in the Milnor short exact sequence).
2. [TVC] implies [CHT] is not known. Our Theorem 6.1. is the other implication :) This result is not due to us but to Danica Kosanovic.
3. [CCH] implies [CVC]. I don't think that's true, what's true is that [CHT] implies [CVC]. In fact, I would say that your three conjectures admit a "homology" version and a "homotopy" version. [GVC] would be the same in both case. Homological [CVC] would be with all Vassiliev invariants and all chord diagrams, homotopical [CVC] would be with additive Vassiliev invariants and indecomposable chord diagrams (with respect to the Hopf algebra structure). Homotopical [TVC] is what you say and homological [TVC] is stating that there is an isomorphism between the order n universal Vassiliev invariant and the 2n-stage of the homological Goodwillie-Weiss spectral sequence (the surjectivity has been shown by Volic in his thesis).

Here is how I would sum up the story. There is the notion of type $\leq n$ invariant and this gives a sequence of quotients of the set of knots $\pi_0(K)/\sim_n$ as you said. The [GVC] is wide open. Independently on that, you might want to compute $\pi_0(K)/\sim_n$. It has been shown by Goussarov that these are finitely generated abelian groups and that there is a surjective map $A_n\to G_n=ker(\pi_0(K)/\sim_n\to \pi_0(K)/\sim_{n-1})$. We know that $A_n$ is the term $E^2_{n+1,n+1}$ in the homotopical Goodwillie-Weiss spectral sequence. Under [TVC], we know that $G_n=E^{\infty}_{n+1,n+1}$, and therefore, if we have [TVC]+[CHT], we know [CVC] (and if you're careful, you will realize that you only need surjectivity of [TVC]+[CHT]). This is the situation that we are at over the rationals and $p$-locally in the range $n\leq p+1$. Finally, if you are not working over a field, it is not clear that $\pi_0(K)/\sim_n$ will be isomorphic to $\oplus_{k\leq n}G_k$ as you might have extension problems. These don't exist in the range that we can compute in our paper but they might exist further.