ScalingStacks

Throughout this section we assume familiarity with planar algebras [Jon99].

We won’t actually spell this out in detail, but one can easily extract from this definition the notion of the lasagna operad (actually a coloured operad, with colours corresponding to links), and that a lasagna algebra is an algebra for that operad. One can of course consider lasagna algebras valued in symmetric monoidal categories other than (bigraded) abelian groups.

The ‘one input ball’ part of a lasagna algebra is essentially equivalent to a functorial invariant of links in 3-spheres: we have an abelian group for each such link, and homomorphisms for cobordisms between them, which compose appropriately. It is not immediately clear that any such functorial invariant extends to a full lasagna algebra, with well-defined operations for multiple input balls. The goal in this section is to show that this is the case for Khovanov–Rozansky homology. In fact, our argument shows that any functorial invariant of links and cobordisms in 3-spheres which satisfies the monoidality property and sweep-around move extends to a lasagna algebra.

In this section, we use the lasagna algebra described above to construct an invariant $\mathcal{S}^{N}_{0}(W;L)$ of smooth oriented 4-manifolds $W$, possibly with a link $L$ in the boundary, valued in bigraded abelian groups. It is akin to the skein modules of 3-manifolds, which can be defined from any ribbon category, except that everything happens one dimension higher. The relationship between this invariant and what we are eventually after is analogous to that between $HH_{0}$ and $HH_{*}$ of an algebra.

The bidegree of $F$ is $\deg(F):=\sum_{i}\deg(v_{i})+(0,(1-N)\chi(\Sigma))$. We will also consider linear combinations of lasagna fillings and impose the relation that lasagna fillings are multilinear in the input labels $v_{i}$. Thus, lasagna fillings of $W$ with boundary $L$ form a bigraded abelian group.

For a 4-ball $W$ with a link $L\in\partial W$, a lasagna filling $F$ is equivalent to the data of a lasagna diagram $D$ together with input labels $v_{i}\in\mathrm{KhR}_{N}(S_{i},L_{i})$. In particular, we can compute the evaluation $\mathrm{KhR}_{N}(F)=\mathrm{KhR}_{N}(D)(v_{i})\in\mathrm{KhR}_{N}(\partial W,L)$.

The relation $\sim$ is homogeneous and, thus, $\mathcal{S}^{N}_{0}(W;L)$ is a bigraded abelian group since the bidegree of a cobordism map $\mathrm{KhR}_{N}(\Sigma)$ is $(0,(1-N)\chi(\Sigma))$ and the Euler characteristic of surfaces is additive under gluing along links.

Having defined the skein module $\mathcal{S}^{N}_{0}(W;L)$, we now proceed to constructing a disklike $4$-category. This will also lead to a more refined invariant, taking the form of a chain complex with 0-th homology $\mathcal{S}^{N}_{0}(W;L)$.

We very briefly recall the key points of the definition of a disklike 4-category, from §6 of [MW12]. A disklike $n$-category ${\mathcal{C}}$ consists of:

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  for each $0\leq k\leq n$, a functor

  $${\mathcal{C}}^{k}:\{\text{$k$-balls and diffeomorphisms}\}\to{\mathsf{Set}}$$

  (and we interpret ${\mathcal{C}}^{k}(X)$ as the set of $k$-morphisms with shape $X$),

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  for each $k-1$-ball $Y$ in the boundary of a $k$-ball $X$, a restriction map ${\mathcal{C}}^{k}(X)\to{\mathcal{C}}^{k-1}(Y)$ (to be more careful, these restriction maps only need to be defined on sufficiently large subsets of ${\mathcal{C}}^{k}(X)$, for example to allow for transversality issues),

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  for each $k$-ball $X$ presented as the gluing of two $k$-balls $X_{1}$ and $X_{2}$ along a common $k-1$-ball $Y$ in their boundaries, a gluing map

  $${\mathcal{C}}^{k}(X_{1})\times_{{\mathcal{C}}^{k-1}(Y)}{\mathcal{C}}^{k}(X_{2})\to{\mathcal{C}}^{k}(X),$$

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  such that these gluing operations are compatible with the action of diffeomorphisms, and associative on the nose,

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  and that two diffeomorphisms of $n$-balls which are isotopic rel boundary act identically,

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  along with some data and axioms concerning identities which we omit here.

(As a reminder, the surprising feature of this definition is that while gluing is required to be strictly associative, this definition actually models fully weak $n$-categories. The key point is that we do not choose canonical models for the shape of a $k$-morphism, and it is up to ‘the end user’ to pick reparametrisations of glued balls back to any standard model balls that they prefer. It is these reparametrisations that are responsible for introducing all the difficult structural isomorphisms of most definitions. This is analogous to the idea of a Moore loop space, which has a strictly associative composition, versus an ordinary loop space, which has a complicated higher associator structure described by Stasheff polyhedra.)

As explained in [MW12], one of the primary examples of a disklike $n$-category is string diagrams for a pivotal traditional $n$-category. This string diagram construction works just as well for a lasagna algebra (which is essentially a pivotal 4-category with trivial 0- and 1-morphisms). Specifically, starting from the lasagna algebra $\mathrm{KhR}_{N}$, we define a disklike $4$-category $\mathsf{KhR}_{N}$ as follows:

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  For $X$ a 0-ball, we define $\mathsf{KhR}_{N}^{0}(X)$ to be a single-element set.

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  For $X$ a 1-ball, we define $\mathsf{KhR}_{N}^{1}(X)$ to be a single-element set.

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  For $X$ a 2-ball, we define $\mathsf{KhR}_{N}^{2}(X)$ to be the set of all configurations of finitely many framed oriented points in $X$.

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  For $X$ a 3-ball, we define $\mathsf{KhR}_{N}^{3}(X)$ to be the set of all framed oriented tangles (not up to isotopy) properly embedded in $X$. If $c$ is a finite configuration of oriented points in $\partial X$, we define $\mathsf{KhR}_{N}^{3}(X;c)$ to be the set of all oriented tangles which restrict to $c$ on $\partial X$.

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  For $X$ a 4-ball, and $L$ a link in $\partial X$, we define $\mathsf{KhR}_{N}^{4}(X;L)$ to be the bigraded abelian group $\mathcal{S}^{N}_{0}(X;L)$ defined above, that is, all lasagna fillings of $X$ which restrict to $L$ on the boundary, modulo relations described above. Recall that by Example 5.6 we know $\mathsf{KhR}_{N}^{4}(X;L)\cong\mathrm{KhR}_{N}(\partial X,L)$.

We define $\mathsf{KhR}_{N}^{4}(X;L)$ to be lasagna fillings modulo relations rather than simply defining it to be $\mathrm{KhR}_{N}(\partial X,L)$ in order to make it easier to define composition below.

We will henceforth drop superscripts and write $\mathsf{KhR}_{N}(X)$ instead of $\mathsf{KhR}_{N}^{k}(X)$.

In dimensions 0 through 3, it is clear that $\mathsf{KhR}_{N}(X)$ is functorial with respect to diffeomorphisms. In dimension 4, it is clear the diffeomorphisms act on lasagna fillings; what remains is to show that the relations we impose are compatible with the action of diffeomorphisms. Specifically, for a diffeomorphism $f$ we must show that if $\mathrm{KhR}_{N}(F)=\mathrm{KhR}_{N}(F^{\prime})$ then $\mathrm{KhR}_{N}(f(F))=\mathrm{KhR}_{N}(f(F^{\prime}))$. This follows from the fact that any diffeomorphism of a 4-ball (rel boundary) is isotopic to the identity away from a small 4-ball in the interior. We can arrange that this small 4-ball is disjoint from $\Sigma$ and the $B_{i}$. The argument is similar to (but simpler than) the argument given in Lemma 4.7.

We must now define gluing (composition) of morphisms. In dimensions 0 through 3 the morphisms are purely geometric and the gluing is defined to be the obvious geometric gluing of submanifolds. In dimension 4, there is again an obvious geometric gluing map of lasagna fillings. We must show that this gluing map is compatible with the relations we impose on fillings. This follows from the operad composition property proved in the previous section.

Finally, the (omitted above) axioms about identities require that we check that 4-ball diffeomorphisms which are supported away from the surface $\Sigma$ act trivially. The diffeomorphism action on lasagna fillings is just moving submanifolds around (and, if the internal balls move, applying the $S^{3}$-functoriality action from the first piece of data for a lasagna algebra to the labels), so a diffeomorphism supported away from the surface and the internal balls does not change a lasagna filling.

Having built a disklike 4-category we immediately obtain an alternative description of the skein module $\mathcal{S}^{N}_{0}(W;L)$ for a link in the boundary of any oriented smooth 4-manifold $W$, as first introduced in §5.2.

This is the construction from [MW12, §6.3], which describes $\mathcal{S}^{N}_{0}(W;L)$ as a colimit, taken over all ways of decomposing a 4-manifold $W$ into a gluing of closed balls (with some regularity conditions on the ways these balls meet). For any such decomposition, we draw compatible links in the boundaries of each of the balls (i.e. if two balls meet along some 3-manifold, the intersections of the two links with that 3-manifold are tangles, and identical, and a similar condition holds for any ball meeting $\partial W$). Then the bigraded abelian group at such a decomposition is the direct sum, over the choices of link labels, of the tensor products of the Khovanov–Rozansky homologies of each link. The arrows in the colimit diagram are ways of coarsening the decomposition by gluing several balls together into a single ball. The gluing maps for a disklike 4-category provide morphisms of bigraded abelian groups. Finally, the skein module invariant $\mathcal{S}^{N}_{0}(W;L)$ associated to $W$ is just the colimit of this diagram.

We will leave it as an exercise to the interested reader to verify that these two constructions actually give the same result!

Our motivation for introducing the disklike 4-category is that the construction of [MW12, §6.3] actually gives much more. Associated to any link $L$ in the boundary of a 4-manifold $W$, we obtain the blob complex (with coefficients in the disklike 4-category $\mathsf{KhR}_{N}$), which we write as ${\mathcal{B}}_{*}(\mathsf{KhR}_{N})(W;L)$. (One approach to the definition of this complex is by replacing the colimit described above with an appropriate homotopy colimit, see [MW12, §7].) This has a new homological grading, unrelated to the internal homological grading from Khovanov-Rozansky homology. The 0-th homology of this complex recovers the bigraded abelian group $\mathcal{S}^{N}_{0}(W;L)$, but the higher blob homology groups, denoted by $\mathcal{S}^{N}_{i}(W;L)$ for $i>0$, potentially carry further information.

Attempting any calculations of this invariant, or of its 0-th homology in either formulation, remains beyond the scope of this paper, and developing appropriate computational tools is an open problem for future work (e.g. [MN20]). One such tool should come from a categorification of the $\mathfrak{gl}_{N}$ skein relation, namely the skein exact triangle for Khovanov–Rozansky chain complexes in ${\mathbb{R}}^{3}$, which induces a long exact sequence on homology groups. For a skein triple of links in the boundary of some interesting 4-manifold $W$ we have every reason to expect that the corresponding sequence on the level of the skein module $\mathcal{S}^{N}_{0}$ is no longer exact. We do, however, obtain long exact sequences on the level of the blob complex, which give rise to a spectral sequence that relates the skein modules $\mathcal{S}^{N}_{0}$ for the three links. In fact, the study of these spectral sequences was the original motivation for the blob complex (however ahistorical this might seem, given the publication dates).