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193 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
193 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
194 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
194 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
195 \stackrel{f_k}{\to} \bc_*(N_0) |
195 \stackrel{f_k}{\to} \bc_*(N_0) |
196 \] |
196 \] |
197 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
197 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
198 \nn{issue: haven't we only defined $\id\ot\alpha_i$ when $\alpha_i$ is closed?} |
198 \nn{need to double check case where $\alpha_i$'s are not closed.} |
199 It is easy to check that the above definition is compatible with the equivalence relations |
199 It is easy to check that the above definition is compatible with the equivalence relations |
200 and also the operad structure. |
200 and also the operad structure. |
201 We can reinterpret the above as a chain map |
201 We can reinterpret the above as a chain map |
202 \[ |
202 \[ |
203 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
203 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |