ScalingStacks

We expect that the above $E_{2}$-page-of-spectral-sequence relationship between $\mathcal{S}_{0}^{2}$ and $H_{RW}^{*,*}$ for $(\natural^{m}(S^{1}\times B^{3}),L)$ generalizes to $(W,L)$ for arbitrary four-manifolds $W$ and links $L$. We give a brief sketch of the reasoning below.

Recall that the Khovanov-Rozansky invariants upon which $\mathcal{S}_{0}^{N}$ is built assign chain complexes to links $L$ and chain maps to link cobordisms, but it is not known that this assignment is functorial (or even well-defined) at the level of complexes. The proof that the homology of these complexes is functorial in the appropriate sense involves showing that certain chain maps are homotopic. If this result could be strengthened to show that certain homotopies between the chain maps are themselves 2nd-order homotopic, and so on for all higher orders, then one could construct a functorial assignment of chain complexes to links in $S^{3}$ and chain maps to link cobordisms.

Let us assume that these conjectured “fully coherent” $\mathfrak{gl}_{N}$ chain complexes for links exist. Then, they can be repackaged as a pivotal $(\infty,4)$-category (with composition maps defined in terms of link cobordisms, as in [27]). This $(\infty,4)$-category can in turn be fed into the machinery of Section 6.3 of [26] (which is closely related to topological chiral homology [23] and factorization homology [1, 2]). The result is a chain-complex-valued invariant $\mathcal{S}_{\infty}^{N}(W,L)$. Its construction involves taking a homotopy colimit of a poset built out of the set of all ball decompositions of $W$ and refinement relationships between these ball decompositions. Concretely, we construct a double complex, with horizontal differentials coming from the $\mathfrak{gl}_{N}$ complexes of links, and vertical differentials coming from the combinatorics of refining ball decompositions of $W$. There is a spectral sequence associated to this double complex, which is itself an invariant of $(W,L)$.

The $E_{2}$ page of this spectral sequence involves first taking homology in the horizontal direction, then computing homology with respect to vertical differentials. It is easy to see that this $E_{2}$ page is exactly the blob homology $\mathcal{S}^{N}_{*}(W;L)$ assigned to $(W,L)$ in [27] (i.e. by taking $\operatorname{KhR}$ homology early instead of working with the $\mathfrak{gl}_{N}$ complex). (In this paper we have focused on blob-degree zero, corresponding to the bottom row of the $E_{2}$ page of the spectral sequence.)

When $W_{1}=\natural^{m}(S^{1}\times B^{3})$ and $N=2$, we expect the total homology of $\mathcal{S}_{\infty}^{N}(W_{1},L)$ to coincide with the Rozansky–Willis invariants. The Hochschild differentials of the previous subsection should be (homotopy equivalent to) special cases of the vertical differentials above.