ScalingStacks

Proof. The map is defined by first considering the direct sum of the gluing morphisms

$$\operatorname{KhR}_{N}(R\cup\bigsqcup_{i}(T_{i}\sqcup\overline{T_{i}}),\mathbbm{k})\{(\textstyle\sum_{i}p_{i})(N-1)\}\to\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$$

from (17). The coinvariants for the actions of $\mathcal{S}_{0}^{N}(B^{3};P_{p_{i}},\mathbbm{k})$ clearly lie in the kernel, so we get an induced map from the indicated quotient to $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$, which we again call the gluing map. It is surjective by Lemma 4.4, so it remains to prove injectivity.

Let $F_{1},F_{2}$ be two equivalent linear combinations of lasagna fillings in $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$, and let $G_{1},G_{2}$ be respective preimages under the gluing map. We want to show that $G_{1}$ and $G_{2}$ are equivalent. Without loss of generality, we may assume that $F_{1}$ and $F_{2}$ are individual lasagna fillings (rather than linear combinations) and that they differ by a single move as in Lemma 2.1 with the relevant input ball fixed and disjoint from the cocores of the 1-handles in $W_{1}$. If $F_{1}$ and $F_{2}$ differ by a replacement inside the fixed input ball or an isotopy supported away from the cocores, then $G_{1}$ and $G_{2}$ are equal in $\operatorname{KhR}_{N}(R\cup\bigsqcup_{i}(T_{i}\sqcup\overline{T_{i}}),\mathbbm{k})$. If $F_{1}$ and $F_{2}$ differ by an isotopy supported in a neighborhood of the cocores, then $G_{1}$ or $G_{2}$ differ by an element of the subspace factored out. Since every isotopy of lasagna fillings can be factored in this way, we get that $G_{1}$ and $G_{2}$ are equivalent. ∎