204 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
204 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
205 \] |
205 \] |
206 The main result of this section is that this chain map extends to the full singular |
206 The main result of this section is that this chain map extends to the full singular |
207 chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. |
207 chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. |
208 |
208 |
209 \begin{prop} |
209 \begin{thm} |
210 \label{prop:deligne} |
210 \label{thm:deligne} |
211 There is a collection of chain maps |
211 There is a collection of chain maps |
212 \[ |
212 \[ |
213 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
213 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
214 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
214 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
215 \] |
215 \] |
216 which satisfy the operad compatibility conditions. |
216 which satisfy the operad compatibility conditions. |
217 On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
217 On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
218 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. |
218 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. |
219 \end{prop} |
219 \end{thm} |
220 |
220 |
221 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
221 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
222 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on |
222 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on |
223 blob cochains. |
223 blob cochains. |
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 |