ScalingStacks

The decomposition  (1) becomes

(2)

$\displaystyle\mathcal{S}_{0}^{N}(W;L)=\bigoplus\limits_{\alpha\in H_{2}^{L}(W;{\mathbb{Z}})}\mathcal{S}_{0}^{N}(W;L,\alpha).$

We will use the decomposition (2) in the case of a general link $L$; when $[L]\neq 0$, we have $H_{2}^{L}(W;{\mathbb{Z}})=\emptyset$ and $\mathcal{S}_{0}^{N}(W;L)=0$.