ScalingStacks

Definition 4.5. Consider a link cobordism $\Sigma\subset W$ in a 4-manifold $W$ diffeomorphic to ${\mathbb{R}}^{3}\times{[0,1]}$. Let $\Sigma_{\mathrm{in}}\subset W_{\mathrm{in}}$ and $\Sigma_{\mathrm{out}}\subset W_{\mathrm{out}}$ denote the boundary links in the incoming and outgoing boundary 3-balls of $W$. Then we define

$$\mathrm{KhR}_{N}(W,\Sigma)\colon\mathrm{KhR}_{N}(W_{\mathrm{in}},\Sigma_{\mathrm{in}})\to\mathrm{KhR}_{N}(W_{\mathrm{out}},\Sigma_{\mathrm{out}})$$

in two steps. First we pick a diffeomorphism $\phi\colon W\to{\mathbb{R}}^{3}\times{[0,1]}$, such that $\phi_{\mathrm{out}}:=\phi|_{W_{\mathrm{out}}}\colon W_{\mathrm{out}}\to{\mathbb{R}}^{3}$ and $\phi_{\mathrm{in}}:=\phi|_{W_{\mathrm{in}}}\colon W_{\mathrm{in}}\to{\mathbb{R}}^{3}$ are such that $\phi_{\mathrm{in}}(\Sigma_{\mathrm{in}})$ and $\phi_{\mathrm{out}}(\Sigma_{\mathrm{out}})$ are both generic and blackboard-framed. Then we declare $\mathrm{KhR}_{N}(W,\Sigma)(\eta)$, for a flat section $\eta\in\mathrm{KhR}_{N}(W_{\mathrm{in}},\Sigma_{\mathrm{in}})$, to be the unique flat section of $\mathrm{KhR}_{N}(W_{\mathrm{out}},\Sigma_{\mathrm{out}})$ with value:

$$\mathrm{KhR}_{N}(W,\Sigma)(\eta)(\phi_{\mathrm{out}})=\mathrm{KhR}_{N}(\phi(\Sigma))(\eta(\phi_{\mathrm{in}}))$$