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42 $$\mathfig{.9}{deligne/intervals}$$ |
42 $$\mathfig{.9}{deligne/intervals}$$ |
43 \caption{A fat graph}\label{delfig1}\end{figure} |
43 \caption{A fat graph}\label{delfig1}\end{figure} |
44 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
44 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
45 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
45 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
46 morphisms of such modules as defined in |
46 morphisms of such modules as defined in |
47 Subsection \ref{ss:module-morphisms}. |
47 \S\ref{ss:module-morphisms}. |
48 |
48 |
49 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
49 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
50 of Figure \ref{delfig1} and ending at the topmost interval. |
50 of Figure \ref{delfig1} and ending at the topmost interval. |
51 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
51 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
52 We remove the bottom interval of the bigon and replace it with the top interval. |
52 We remove the bottom interval of the bigon and replace it with the top interval. |
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 \S\ref{sec:evaluation}. |
219 \end{thm} |
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. |