ScalingStacks

Definition 4.13. Consider a link cobordism $\Sigma\subset W$ in a 4-manifold $W$ diffeomorphic to $S^{3}\times{[0,1]}$. 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-spheres of $W$. Now we define

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

by first choosing a path $p_{t}\subset W\setminus\Sigma$ from $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ to $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$. Then we have $W\setminus\{(p_{t},t)\}_{t\in{[0,1]}}\cong{\mathbb{R}}^{3}\times{[0,1]}$ and 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

$$\mathrm{KhR}_{N}(W,\Sigma)(\eta)(p_{\mathrm{out}})=\mathrm{KhR}_{N}(W\setminus\{(p_{t},t)\}_{t\in{[0,1]}},\Sigma)(\eta(p_{\mathrm{in}}))$$