Example 5.6 (0NBG) · ScalingStacks

0NBG

Example 5.6. If $W$ is a standard 4-ball with $L\subset\partial W=S^{3}$ , then the evaluation of lasagna fillings induces an isomorphism $\ev\colon\mathcal{S}^{N}_{0}(W;L)\cong\mathrm{KhR}_{N}(S^{3},L)$ . In other words, the above complicated quotient yields the usual $\mathrm{KhR}_{N}(S^{3},L)$ in this case.

0NBH

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. ∎