ScalingStacks

Observe that, by excision, $H_{3}(W^{\prime},W\cup(Z\cup L);{\mathbb{Z}})\cong H_{3}(C,C^{\prime};{\mathbb{Z}})=0$. From here we obtain an exact sequence

(8)

$$0\to H_{2}(W^{\prime},L;{\mathbb{Z}})\to H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})\to H_{2}(C,C^{\prime};{\mathbb{Z}}).$$

Thus, an element in $H_{2}^{L}(W^{\prime};{\mathbb{Z}})\subseteq H_{2}(W^{\prime},L;{\mathbb{Z}})$ can be identified with its image in $H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})$, which we write as a pair $(\alpha,\eta)$.