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.