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169 In this case the surgery cylinder is just a single mapping cylinder. |
169 In this case the surgery cylinder is just a single mapping cylinder. |
170 |
170 |
171 \medskip |
171 \medskip |
172 |
172 |
173 Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$. |
173 Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$. |
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174 As usual, fix a system of field $\cF$ and let $\bc_*$ denote the blob complex construction based on $\cF$. |
174 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
175 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
175 as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
176 as modules of the $A_\infty$ 1-category $\bc_*(E_i)$ (see \S\ref{ss:module-morphisms}). |
176 We define a map |
177 We will define a map |
177 \[ |
178 \[ |
178 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
179 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
179 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
180 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
180 \] |
181 \] |
181 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define |
182 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define |