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224 As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes |
224 As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes |
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 homeomorphisms spaces, modulo some relations. |
229 union of products of homeomorphism spaces, modulo some relations. |
230 By \ref{CHprop}, |
230 By Proposition \ref{CHprop} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
231 \nn{...} |
231 a chain map |
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232 \[ |
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233 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
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234 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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235 \] |
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236 It suffices to show that the above maps are compatible with the relations whereby |
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237 $FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
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238 This in turn follows easily from the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
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239 |
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240 \nn{should add some detail to above} |
232 \end{proof} |
241 \end{proof} |
233 |
242 |
234 \nn{maybe point out that even for $n=1$ there's something new here.} |
243 \nn{maybe point out that even for $n=1$ there's something new here.} |