176 We define a map |
176 We define a map |
177 \[ |
177 \[ |
178 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
178 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
179 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
179 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
180 \] |
180 \] |
181 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition |
181 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define |
|
182 $p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition |
182 \[ |
183 \[ |
183 \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
184 \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
184 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
185 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
185 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
186 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
186 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
187 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
199 |
200 |
200 \begin{thm} |
201 \begin{thm} |
201 \label{thm:deligne} |
202 \label{thm:deligne} |
202 There is a collection of chain maps |
203 There is a collection of chain maps |
203 \[ |
204 \[ |
204 C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
205 C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
205 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
206 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
206 \] |
207 \] |
207 which satisfy the operad compatibility conditions. |
208 which satisfy the operad compatibility conditions. |
208 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
209 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
209 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. |
210 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. |
214 blob cochains. |
215 blob cochains. |
215 As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
216 As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
216 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |
217 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |
217 |
218 |
218 \begin{proof} |
219 \begin{proof} |
219 As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
220 As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint |
220 union of products of homeomorphism spaces, modulo some relations. |
221 union of products of homeomorphism spaces, modulo some relations. |
221 By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
222 By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
222 a chain map |
223 a chain map |
223 \[ |
224 \[ |
224 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
225 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
225 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
226 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
226 \] |
227 \] |
227 It suffices to show that the above maps are compatible with the relations whereby |
228 It suffices to show that the above maps are compatible with the relations whereby |
228 $SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
229 $SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s. |
229 This in turn follows easily from the fact that |
230 This in turn follows easily from the fact that |
230 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
231 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
231 %\nn{should add some detail to above} |
232 %\nn{should add some detail to above} |
232 \end{proof} |
233 \end{proof} |
233 |
234 |