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225 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. |
225 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. |
226 |
226 |
227 \begin{proof} |
227 \begin{proof} |
228 As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
228 As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
229 union of products of homeomorphism spaces, modulo some relations. |
229 union of products of homeomorphism spaces, modulo some relations. |
230 By Proposition \ref{CHprop} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
230 By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
231 a chain map |
231 a chain map |
232 \[ |
232 \[ |
233 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
233 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
234 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
234 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
235 \] |
235 \] |