ScalingStacks

Remark 3.9. Example 3.8 gives an alternate formula for 3-handle attachments. Let us go back to the general setting in this section, with a 3-handle attached to an arbitrary four-manifold $W$ along a sphere $S$ to produce $W^{\prime}$, and a framed link $L\subseteq\partial W$ away from $S$. Observe that $\mathcal{S}_{0}^{N}(W,L)$ is naturally a module over the algebra $\mathcal{S}_{0}^{N}(S^{2}\times D^{2};\emptyset)$, with the module action being given by attaching fillings in a neighborhood of the sphere $S$. It follows from the definitions that

$$\mathcal{S}_{0}^{N}(W^{\prime};L^{\prime})\cong\mathcal{S}_{0}^{N}(W;L)\otimes_{\mathcal{S}_{0}^{N}(S^{2}\times D^{2};\emptyset)}\mathcal{S}_{0}^{N}(B^{3}\times I;\emptyset).$$

Here, the algebra $\mathcal{S}_{0}^{N}(S^{2}\times D^{2};\emptyset)$ is the free polynomial ring in $A_{1},\dots,A_{N-1},A_{0},A_{0}^{-1}$ and $\mathcal{S}_{0}^{N}(B^{3}\times I;\emptyset)=\mathcal{S}_{0}^{N}(B^{4})$ is ${\mathbb{Z}}$ as a module over that algebra, where $A_{0}$ acts by $1$ and the other $A_{i}$ by $0$. We conclude that

$$\mathcal{S}_{0}^{N}(W^{\prime};L^{\prime})\cong\mathcal{S}_{0}^{N}(W;L)/(A_{0}-1,A_{1},\dots,A_{N}).$$