ScalingStacks

Proof. It follows from Theorem 5.2 that equivalent lasagna fillings of $W$ have equal evaluation. Thus, we get a well-defined homomorphism $\ev\colon\mathcal{S}^{N}_{0}(W;L)\to\mathrm{KhR}_{N}(S^{3},L)$, which is surjective since any homogeneous $v\in\mathrm{KhR}_{N}(S^{3},L)$ appears as the image of a radial lasagna filling $F_{v}$. Similarly, if two lasagna fillings $F_{1}$ and $F_{2}$ have equal evaluation $v\in\mathrm{KhR}_{N}(S^{3},L)$, then we observe $F_{1}\sim F_{v}\sim F_{2}$, and so $\ev$ is injective. ∎