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]). ∎

Proof. The first isomorphism is given by a 1-handle attachment to $(B^{4},\emptyset)$ as in Lemma 4.1. The second isomorphism can be proved similarly: Let $F$ be a lasagna filling of $S^{1}\times S^{3}$ and consider its intersection with a fiber $\{x\}\times S^{3}$. Up to a small isotopy, we may assume that the filling $F$ intersects $\{x\}\times S^{3}$ transversely (in lasagna sheet, not in input balls) and disjointly from $\{x\}\times\{\text{north pole}\}$. Then for small $\epsilon>0$, the intersection $F\cap[x-\epsilon,x+\epsilon]\times(S^{3}\setminus\text{north pole})$ is an identity cobordism on a link $K$. We replace this by a sum over pairs of input balls labelled with basis and dual basis elements of $\operatorname{KhR}_{N}(K)$ respectively. The resulting closed lasagna filling is supported in a single $B^{4}$ and can, thus, be identified with a scalar multiple of the empty filling. ∎