ScalingStacks

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 $\mathcal{S}^{N}_{0}(W;L)$.