ScalingStacks

We first consider the case when $L$ is disjoint from the cocores of the 1-handles. Up to a small isotopy, we may even assume that $L$ is disjoint from the entire boundary of the added 1-handles, i.e. that $L\subset\partial W$. As in Proposition 3.4, the corresponding invariants are related by a canonical map and we have:

Consider the process of cutting a lasagna filling $F$ of $W_{1}=\natural^{m}(S^{1}\times B^{3})$ with boundary $L$ along the cocores $C_{i}\cong\mathrm{pt}\times B^{3}$ of the 1-handles for $1\leq i\leq m$. Let us assume that the lasagna sheet $\Sigma$ of $F$ intersects the cocores transversely in tangles $T_{i}:=\Sigma\cap C_{i}$. In particular, the link $L$ intersects the belt spheres $S_{i}:=\partial C_{i}$ geometrically in $2p_{i}$ points, the boundary points of the tangle $T_{i}$. The algebraic intersection numbers are all zero, since $L$ is null-homologous, as witnessed by $F$. In this way, we obtain a lasagna filling $\mathrm{cut}(F)$ of $W_{1}\setminus\bigsqcup_{i}n(C_{i})\cong B^{4}$ with boundary link

The latter is obtained by cutting $L$ open at the $2p_{i}$-tuples of boundary points and inserting copies of the tangles $T_{i}$ and $\overline{T_{i}}$, schematically:

Of course, the procedure of cutting lasagna fillings does not describe a well-defined map on the level of $\mathcal{S}_{0}^{N}$ since it does not respect the skein relations. Instead we consider the reverse operation.

The process of gluing a lasagna filling works as follows. Let $F^{\prime}$ be a lasagna filling of $B^{4}$ with boundary link $L_{T}$ as above; i.e., inside $S^{3}=\partial B^{4}$ we have $m$ pairs of embedded 3-balls $B_{i}\cup\overline{B_{i}}$, such that $L_{T}\cap B_{i}=T_{i}$ and $L_{T}\cap\overline{B_{i}}=\overline{T}_{i}$ for $1\leq i\leq m$. Denote the numbers of boundary points by $2p_{i}:=|\partial T_{i}|$. Now we attach $m$ 1-handles with core-parallel lasagna sheets $I\times T_{i}\subset I\times B^{3}$ along the $B_{i}\cup\overline{B_{i}}\cong S^{0}\times B^{3}$ to obtain a lasagna filling of $W_{1}$ with boundary $L$. Since the relations in $\mathcal{S}_{0}^{N}$ are local, this induces a map:

The grading shift is there to compensate the change in Euler characteristic of the surfaces in lasagna fillings upon gluing.

It follows that the gluing maps from (17) assemble to a surjective map from a direct sum of shifts of $\mathcal{S}_{0}^{N}(B^{4};L_{T},\mathbbm{k})$ to $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$. Here, the sum is indexed by all ways of writing $L$ as a contraction of links $L_{T}$ obtained by drilling out pairs of tangles $T_{i}\cup\overline{T_{i}}$ and resealing the boundary points across the 1-handles. It remains to describe the kernel.

The statement of Lemma 4.6 can be paraphrased as: the choice of a $3$-ball with point configuration $P_{p}$ in $\partial W$ equips $\mathcal{S}_{0}^{N}(W;-,\mathbbm{k}):=\bigoplus_{L}\mathcal{S}_{0}^{N}(W,L,\mathbbm{k})$ with the structure of a bigraded module for the category $\mathcal{S}_{0}^{N}(B^{3};P_{p})$. (Here the direct sum is taken over all links $L$ that intersect the boundary of the chosen $3$-ball in the fixed configuration $P_{p}$.)

Theorem 4.7 can also be summarized by saying that $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ is computed by the zeroth Hochschild homology of a tensor product of $3$-ball categories, namely one for each handle, with coefficients in a bimodule associated to the tangle $R$ that results from $L$ by cutting open along the belt spheres. We will discuss the details of this perspective in a special case in Section 4.3.

Finally we comment on the functoriality of the 1-handle formula from Theorem 4.7. We have seen that $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ for $W_{1}=\natural^{m}(S^{1}\times B^{3})$ is a colimit of link homologies for links in $S^{3}$, which result from cutting $L$ along belt spheres and inserting pairs of tangles. Now consider a link cobordism $S\subset\partial W_{1}\times I=:Z$ from $L\subset W_{1}$ to $L^{\prime}\subset W_{1}^{\prime}$ where $W_{1}^{\prime}=W_{1}\cup Z$. We claim that the induced map

can also be expressed in terms of cobordism maps between links in $S^{3}$. Recall that the cobordism map $\Psi_{Z;S}$ sends a lasagna filling $F$ of $W_{1}$ to the composite lasagna filling $F\cup S$ of $W_{1}\cup Z$. In a generic situation, cutting the cocores has the following local model. Here we display the filling $F$ in the inner tube and $S$ in the outer, spherical shell.

Let $S_{i}$ denote the tangle in $S^{2}\times I$ that occurs as the intersection of $S$ with the $i$th cocore and $R^{\prime}\subset S^{3}\setminus\bigsqcup_{i}(B_{i}\cup\overline{B_{i}})$ the tangle obtained from $L^{\prime}$ by cutting open along the belt spheres of $W_{1}^{\prime}$. Denote by $2p^{\prime}_{i}=|\partial S_{i}|-2p_{i}$ the number of outer boundary point of $S_{i}$. Then the cobordism $\Sigma$ obtained from $S$ by cutting along the annuli, which are the intersection of $Z$ with the cocores of 1-handles in $W_{1}^{\prime}$, induces a cobordism map:

We claim that these components describe $\Psi_{Z;S}$ in terms of the colimit formulas (left-hand sides) from Theorem 4.7. To see this we first observe that the unequal grading shifts guarantee that the components have the same degree as $\Psi_{Z;S}$ (we have $\chi(\Sigma)=\chi(S)+\sum_{i}(p_{i}+p^{\prime}_{i})$ and $\Sigma$ is glued to $\mathrm{cut}(F)$ along $p_{i}$ interval segments). Next we observe that after composing with the projection-inclusion into the colimit formula for $\mathcal{S}_{0}^{N}(W_{1}^{\prime};L^{\prime},\mathbbm{k})$, the resulting map no longer depends on the chosen location of cocores to cut. Moreover, the subspace factored out in the colimit formula for $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ is annihilated by the map thus defined. Thus the components described above define a map $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})\to\mathcal{S}_{0}^{N}(W_{1}^{\prime};L^{\prime},\mathbbm{k})$, and by construction this agrees with $\Psi_{Z;S}$.

Recall the following definition, from e.g. [4].

If $\mathcal{C}$ as in Definition 4.9 is not just enriched in $K$-modules, but $M$-graded $K$-modules for some monoid $M$, then $\mathrm{HH}_{0}(\mathcal{C})$ inherits the structure of an $M$-graded $K$-module. The following is now an immediate consequence of Theorem 4.7 and the Definitions 4.5 and 4.9.

We now recall some facts about the zeroth Hochschild homology, which we will use to show that the 1-handle formula may compute vector spaces which are not locally finite-dimensional.

In a slight reformulation of the functoriality results from [8], the tangle invariant underlying the $\mathfrak{gl}_{N}$ link homology over $\mathbbm{k}$ can be described as a 2-functor:

We now briefly explain the relevant algebraic structures here.

• •

  As in [27, Definition 6.1] one defines a category $\mathbf{TD}$ of tangle diagrams, whose objects are finite words in the alphabet $\{\uparrow,\downarrow\}$ (which encode possible sequences of oriented boundary points for tangles) and whose morphisms are finite words in generating morphisms $\{\mathrm{cup}_{i},\mathrm{cap}_{i},\mathrm{crossing}_{i},\mathrm{crossing}_{i}^{-1}\}$ (where the index $i$ specifies the strands participating in the generator), that are admissible in the sense that the composite describes a tangle diagram. The composition is concatenation of words. For details see [27, Definition 6.1].

• •

  $\boldsymbol{\mathrm{Tang}}$ is a $2$-category whose objects and $1$-morphisms are as in $\mathbf{TD}$. The $2$-morphisms are the framed, oriented tangle cobordisms in $[0,1]^{4}$ between standard lifts of tangle diagrams to actual tangles in $[0,1]^{3}$, considered up to isotopy rel boundary.

• •

  $\boldsymbol{\mathrm{Foam}}_{N}$ is a (monoidal) $2$-category, enriched at the level of $2$-morphism spaces in $\mathbbm{k}$-vector spaces and equipped with grading shift functors on $1$-morphisms. It has the same objects²² 2 More generally, one can consider labelled oriented points as objects in $\boldsymbol{\mathrm{Foam}}_{N}$, but we will not need labels other than $1$. as $\boldsymbol{\mathrm{Tang}}$. The $1$-morphisms are (formal direct sums of grading shifts of) $\mathfrak{gl}_{N}$ webs embedded in $[0,1]^{2}$ and the $2$-morphisms are (matrices with entries given by) $\mathbbm{k}$-linear combinations of $\mathfrak{gl}_{N}$ foams embedded in $[0,1]^{3}$, modulo certain local relations. For details see [8].

• •

  $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$ is the (monoidal) $2$-category that is obtained from $\boldsymbol{\mathrm{Foam}}_{N}$ by replacing its $\mathbbm{k}$-linear $\operatorname{Hom}$-categories by the corresponding dg categories. This means it has the same objects, but the $1$-morphisms are now chain complexes formed from $1$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$, where the differentials are given by $2$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$. The $2$-morphisms spaces are chain complexes of homologically homogeneous and quantum grading-preserving maps, spanned by $2$-morphisms from $\boldsymbol{\mathrm{Foam}}_{N}$ (not necessarily chain maps). The differential on $2$-morphisms is the usual supercommutator with respect to the differential on the source- and target complexes. With respect to this differential the zero cycles are exactly the classical chain maps. There is also an enriched $2$-hom in $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$, which is assembled from $2$-homs between objects shifted in quantum grading.

• •

  $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$ is the cohomology category of $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$. It has the same objects and $1$-morphisms, but the $2$-morphism spaces are now graded $\mathbbm{k}$-modules obtained by taking cohomology. The zeroth cohomology $H^{0}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$ is also called the homotopy category; its $2$-morphisms are chain maps up to homotopy.

  In the following we will also consider enriched $2$-homs. For objects $s,t$ and $1$-morphisms $A,B\colon s\to t$ we define the bigraded $\mathbbm{k}$-modules:

  (21)

  $$H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(A,B):=\bigoplus_{k\in{\mathbb{Z}}}\operatorname{Hom}_{H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})}(A\{k\},B)$$

  Here one grading, the quantum grading, is given by the displayed direct sum, while the other grading, the homological grading, is already internal to $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$. Using the grading shift automorphisms, these enriched $2$-homs admit composition maps and thus assemble into a bigraded $\mathbbm{k}$-linear enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(s,t)$ whose objects are the $1$-morphisms from $s$ to $t$.

• •

  The functor $\left\llbracket-\right\rrbracket$ is the identity on objects. On $1$-morphisms it sends a tangle diagram to a chain complex of webs and foams in the way that is usual for $\mathfrak{gl}_{N}$ link homology, and $2$-morphisms, i.e. isotopy classes of tangle cobordisms are sent to the corresponding homotopy classes of chain maps as specified in the functoriality proof in [8].

We recall from [27, Section 6] that the tangle invariant corresponding to the $\mathfrak{gl}_{N}$ link homology can be organized into a braided monoidal $2$-category. Here we give a similar construction of this category $\boldsymbol{\mathrm{T}}_{N}$ (which was denoted $\mathbf{KhR}_{N}$ in[27]) by replacing the top morphism layer of $\boldsymbol{\mathrm{Tang}}$:

• •

  objects are sequences of tangle endpoints, as in $\mathbf{TD}$ and $\boldsymbol{\mathrm{Tang}}$,

• •

  1-morphisms consist of Morse data for tangles, as in $\mathbf{TD}$ and $\boldsymbol{\mathrm{Tang}}$,

• •

  2-morphisms between tangles $S$ and $T$ with equal source and target objects are the bigraded $\mathbbm{k}$-modules computed as the enriched 2-hom $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(\left\llbracket S\right\rrbracket,\left\llbracket T\right\rrbracket)$ from (21) between the $\mathfrak{gl}_{N}$ chain complexes of the tangles.

As an important special case, one gets for a framed, oriented link $L$:

Moreover, if $T$ and $S$ are framed, oriented tangles with endpoints identified, so that we can form the link $T\cup\overline{S}$, then we set $2p=|\partial S|=|\partial T|$ and have:

Given a $3$-ball $B^{3}$ with a set $P_{p}$ of $2p$ framed, co-oriented points in the boundary, together with a suitable identification of $(B^{3},P_{p})$ with $([0,1]^{3},s\cup t)$, we associate to it the morphism category $\boldsymbol{\mathrm{T}}_{N}(s,t)$, whose objects are tangles from $s$ to $t$. By construction, $\boldsymbol{\mathrm{T}}_{N}(s,t)$ is equivalent to $\mathcal{S}_{0}^{N}(B^{3};P_{p})$ from Definition 4.5. Moreover, $\boldsymbol{\mathrm{T}}_{N}(s,t)$ can be considered as a full subcategory of the bigraded enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(s,t)$.

Here we consider the categories from Section 4.3 in the special case when the source and target objects consist of a single point $s=t=\{*\}$. In this case, the corresponding morphism category in $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$ is known to be equivalent to the dg category of complexes of free graded $R_{N}:=\mathbbm{k}[X]/(X^{N})$-modules; see e.g. [32, Lemma 3.35] for an argument in an equivalent setting. We record this equivalence and its consequence on the level of homology:

Here $R_{N}\mathrm{-mod}^{\mathrm{gr.fr.}}$ refers to the category of finitely-generated graded free $R_{N}$-modules and $\operatorname{Ch}_{\mathrm{dg}}(\mathcal{C})$ refers to the dg category of bounded chain complexes over an additive category $\mathcal{C}$. Again we will use a superscript $*$ to refer to the corresponding enriched morphism spaces, computed via the ordinary morphism spaces between shifts of objects as in (21).

Now we specialize to $N=2$ and classify the indecomposable objects. Setting $R:=R_{2}=\mathbbm{k}[X]/(X^{2})$, the isomorphism classes of indecomposable objects (up to shifts in quantum and homological degrees) in $H^{\bullet}(\operatorname{Ch}_{\mathrm{dg}}(R\mathrm{-mod}^{\mathrm{gr.fr.}}))$ are of the form:

for $k\geq 0$; see [19, Section 3].

Next we compute the zeroth Hochschild homology of $H^{\bullet}(\operatorname{Ch}_{\mathrm{dg}}(R\mathrm{-mod}^{\mathrm{gr.fr.}}))$. In principle, there are two possible versions: using the ordinary or the enriched hom; see [5, Section 2.4]. In the case of the ordinary hom, we would obtain a ${\mathbb{Z}}$-graded (namely homologically graded) $\mathbbm{k}[q^{\pm 1}]$-module, where $q$ records the action of the auto-equivalence provided by the shift in quantum grading. We will, however, use the enriched hom (indicated by the superscript $*$) to consider the morphism spaces as bigraded. In doing so, one obtains translation isomorphisms, which identify an object with all its gradings shifts. More specifically, between an object and its shift, the identity now represents an isomorphism of degree specified by the shift. The zeroth Hochschild homology of the resulting category carries the structure of a bigraded $\mathbbm{k}$-vector space, since the endomorphism $q$ now acts as the identity.

Note that the bigraded zeroth Hochschild homology of $H^{\bullet}(\operatorname{Ch}_{\mathrm{dg}}(R\mathrm{-mod}^{\mathrm{gr.fr.}}))^{*}$ is not locally finite-dimensional! It is of countable dimension in bidegree $(0,0)$ with a basis given by $[\operatorname{Id}_{C_{l}}]$ for $l\geq 0$. Nevertheless, we have:

We claim that the $3$-ball categories with $2p\geq 4$ points have zeroth Hochschild homologies that are no longer locally finite-dimensional. Again we restrict to the case of $N=2$ and work over a perfect field $\mathbbm{k}$. Our strategy is to give a lower bound for the dimension of the zeroth Hochschild homology in terms of the split Grothendieck group. We briefly recall the relevant notions and results.

The following is clear from the definition:

Using these tools, we can now prove: