ScalingStacks

Proof. The proof is a straightforward generalization of the proof of [25, Theorem 1.4], which deals with boundary connected sums. The map $i_{*}$ is induced by the map sending lasagna fillings of $(W,L)$ to lasagna fillings of $(W^{\prime},L)$ along the embedding $i$. The inverse is given on lasagna fillings $F$ in $(W^{\prime},L)$ by looking at their intersection with a neighborhood of the cocores of all 1-handles. Up to a small isotopy, each such intersection is an identity cobordism on a link $K\subset B^{3}$. The inverse map is given by replacing it by a sum of pairs of input balls, labelled by basis and dual basis elements of $\operatorname{KhR}_{N}(K)$ respectively. The resulting linear combination of fillings can be isotoped into $W$, and is equivalent to the original filling according to the neck-cutting lemma (Lemma 7.2 in [25]). ∎