ScalingStacks

Let us consider two four-manifolds $W$ and $Z$ that have some part $Y$ of their boundaries in common, as follows:

$$\partial W=Y\amalg Y_{0},\ \ \partial Z=(-Y)\amalg Y_{1},$$

where $\amalg$ denotes disjoint union. We can glue $W$ and $Z$ along $Y$ to form a new four-manifold $W\cup Z$ with boundary $Y_{0}\amalg Y_{1}$. Suppose we are also given links $L_{0}\subset Y_{0}$, $L_{1}\subset Y_{1}$ and $L\subset Y$. Let $\overline{L}\subset-Y$ denote the mirror reverse of $L$. Then, we have a map

(3)

$$\Psi:\mathcal{S}_{0}^{N}(W;L\cup L_{0})\otimes\mathcal{S}_{0}^{N}(Z;\overline{L}\cup L_{1})\to\mathcal{S}_{0}^{N}(W\cup Z;L_{0}\cup L_{1})$$

obtained by gluing lasagna fillings along $L$:

$$[F]\otimes[G]\mapsto[F\cup G].$$

It is easy to see that if two lasagna fillings $F_{1}$ and $F_{2}$ are equivalent in $W$, and $G_{1}$ and $G_{2}$ are equivalent in $Z$, then $F_{1}\cup G_{1}$ and $F_{2}\cup G_{2}$ are equivalent in $W\cup Z$, so (3) is well-defined.

Starting from here, we see that skein lasagna modules are functorial under inclusions, in the following sense. We consider the case when $Y_{0}=\emptyset$, and we fix a lasagna filling $G$ of $Z$ with boundary $\overline{L}\cup L_{1}$. We can think of $Z$ as a cobordism from $Y=\partial W$ to $Y_{1}$. Then, there is an induced cobordism map

(4)

$$\Psi_{Z;G}=\Psi(\cdot\otimes[G]):\mathcal{S}_{0}^{N}(W;L)\to\mathcal{S}_{0}^{N}(W\cup Z;L_{1}).$$

Observe that the maps (4) behave well with respect to compositions:

(5)

$$\Psi_{Z^{\prime};G^{\prime}}\circ\Psi_{Z;G}=\Psi_{Z\cup Z^{\prime};G\cup G^{\prime}}.$$

Furthermore, in terms of the decompositions (2), given $\alpha\in H^{L}_{2}(W;{\mathbb{Z}})$, by attaching to it the class of $G$ in $H_{2}^{L\cup L_{1}}(Z;{\mathbb{Z}})$ we get a class $\alpha_{1}\in H_{2}^{L_{1}}(W\cup Z;{\mathbb{Z}})$. Then, $\Psi_{Z;G}$ maps $\mathcal{S}_{0}^{N}(W;L,\alpha)$ to $\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1})$. We let

(6)

$$\Psi_{Z;G,\alpha}:\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1})$$

denote the restriction of $\Psi_{Z;G}$.

When the lasagna filling $G$ consists of a surface $S$ (an embedded cobordism $S\subset Z$ from $L$ to $L_{1}$) with no input balls, we will simply write $\Psi_{Z;S,\alpha}$ for $\Psi_{Z;G,\alpha}$. Furthermore, we could decorate $S$ with $n$ dots at a chosen location, for $0\leq n\leq N-1$, as usual in $\mathfrak{gl}_{N}$ foams; cf. [27, Example 2.3]. This corresponds to constructing a lasagna filling $S(n\bullet)$ with $n$ input balls intersecting $S$ along unknots, each decorated with the generator

$$X\in\operatorname{KhR}_{N}(U)\cong{\mathbb{Z}}[X]/(X^{N}).$$

(This filling is equivalent to one where we consider a single input ball intersecting $S$ in an unknot, decorated with $X^{n}$.) When the chosen location of the dot placement is clear from the context, then we denote the corresponding map by

(7)

$$\Psi_{Z;S(n\bullet),\alpha}:\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1}).$$