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203 There is a collection of chain maps |
203 There is a collection of chain maps |
204 \[ |
204 \[ |
205 C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
205 C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
206 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
206 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
207 \] |
207 \] |
208 which satisfy the operad compatibility conditions. |
208 which satisfy the operad compatibility conditions, up to coherent homotopy. |
209 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. |
210 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}. |
211 \end{thm} |
211 \end{thm} |
212 |
212 |
213 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
213 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
226 \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)) . |
227 \] |
227 \] |
228 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 |
229 $SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s. |
229 $SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s. |
230 This in turn follows easily from the fact that |
230 This in turn follows easily from the fact that |
231 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 |
|
232 (up to coherent homotopy). |
232 %\nn{should add some detail to above} |
233 %\nn{should add some detail to above} |
233 \end{proof} |
234 \end{proof} |
234 |
235 |
235 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
236 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
236 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |
237 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |