ScalingStacks

Proof. We first show that $\Psi_{Z;A,\alpha^{\prime}}$ vanishes on the image of $f$, that is,

$$\Psi_{Z;A,\alpha^{\prime}}\circ\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}=\Psi_{Z;A,\alpha^{\prime}}\circ\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}.$$

Indeed, from the composition law (5) we see that the left hand side is associated to the surface cobordism $\Delta_{+}\cup A$ and the right hand side to $\Delta_{-}\cup A$. However, inside the 3-handle $Z$, the sphere $S$ gets filled with a core $B^{3}$, and therefore $\Delta_{+}$ and $\Delta_{-}$ are isotopic rel boundary. It follows that the two cobordism maps are the same.

Therefore, $\Psi_{Z;A,\alpha}$ factors through a map

$$\Phi\colon\Bigl(\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W,L,\alpha)\Bigr)/\operatorname{im}(f)\to\mathcal{S}_{0}^{N}(W^{\prime},L^{\prime},\alpha^{\prime}).$$

We need to prove that $\Phi$ is bijective. For this, we construct its inverse $\Phi^{-1}$. Given a lasagna filling $F^{\prime}$ of $W^{\prime}$ with boundary $L^{\prime}$, observe that the cocore of the 3-handle $Z$ is one-dimensional, and therefore we can isotope $F^{\prime}$ to be disjoint from this cocore; after this, we can push it into $W$, to obtain a lasagna filling there, called $F$, with boundary $L$. We set

$$\Phi^{-1}[F^{\prime}]=[F].$$

To see that $\Phi^{-1}$ is well-defined, we need to check that if two lasagna fillings $F^{\prime}_{0}$ and $F^{\prime}_{1}$ are equivalent in $W$, then the corresponding fillings $F_{0}$ and $F_{1}$ differ (up to equivalences in $W$) by an element of $\operatorname{im}(f)$. We use Lemma 2.1, in which we fix balls $R_{i}\subset W$ away from the 3-handle, and consider the equivalences listed in the lemma (with the ball replacements happening in $R_{i}$). Then, the equivalences in $W^{\prime}$ give rise to equivalences in $W$, with one exception: an isotopy of the surfaces may intersect the one-dimensional cocore of $Z$ (which is an interval). Generically, this happens in a finite set of points, each point at a different time during the isotopy. Every time the isotopy meets the cocore, the corresponding surfaces in $W$ differ by replacing a hemisphere of $S$ (with boundary some closed curve $\gamma$) with its complement in $S$. Up to an isotopy supported near $S$, we can assume that $\gamma$ is the equator $J$ with its chosen orientation. (For example, if $\gamma$ is $J$ with the opposite orientation, we can rotate it by $\pi$ about a transverse axis to get $J$ with the original orientation.) Then, the hemispheres being interchanged are $\Delta_{+}$ and $\Delta_{-}$ and hence the classes of $F_{0}$ and $F_{1}$ differ by an element in the image of $f$.

This shows that $\Phi^{-1}$ is well-defined, and its definition makes it clear that it is an inverse to $\Phi$. It follows that $\Phi$ is bijective, and the conclusions follow. ∎