ScalingStacks

Furthermore, in terms of the decompositions (2), given $\alpha\in H^{L}_{2}(W;{\mathbb{Z}})$, by attaching to it the class of $G$ in $H_{2}^{L\cup L_{1}}(Z;{\mathbb{Z}})$ we get a class $\alpha_{1}\in H_{2}^{L_{1}}(W\cup Z;{\mathbb{Z}})$. Then, $\Psi_{Z;G}$ maps $\mathcal{S}_{0}^{N}(W;L,\alpha)$ to $\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1})$. We let

(6)

$$\Psi_{Z;G,\alpha}:\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W\cup Z;L_{1},\alpha_{1})$$

denote the restriction of $\Psi_{Z;G}$.