30 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\ |
30 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\ |
31 & \hspace{-5em} \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
31 & \hspace{-5em} \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
32 \end{eqnarray*} |
32 \end{eqnarray*} |
33 See Figure \ref{delfig1}. |
33 See Figure \ref{delfig1}. |
34 \begin{figure}[!ht] |
34 \begin{figure}[!ht] |
35 $$\mathfig{.9}{tempkw/delfig1}$$ |
35 $$\mathfig{.9}{deligne/intervals}$$ |
36 \caption{A fat graph}\label{delfig1}\end{figure} |
36 \caption{A fat graph}\label{delfig1}\end{figure} |
37 |
37 |
38 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
38 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
39 of Figure \ref{delfig1} and ending at the topmost interval. |
39 of Figure \ref{delfig1} and ending at the topmost interval. |
40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
52 |
52 |
53 It should now be clear how to generalize this to higher dimensions. |
53 It should now be clear how to generalize this to higher dimensions. |
54 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
54 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
55 involved were 1-dimensional. |
55 involved were 1-dimensional. |
56 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries |
56 Thus we can define a $n$-dimensional fat graph to be a sequence of general surgeries |
57 on an $n$-manifold. |
57 on an $n$-manifold. |
58 More specifically, |
58 More specifically, |
59 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
59 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
60 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
60 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
61 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
61 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
62 (See Figure \ref{delfig2}.) |
62 (See Figure \ref{delfig2}.) |
63 \begin{figure}[!ht] |
63 \begin{figure}[!ht] |
64 $$\mathfig{.9}{tempkw/delfig2}$$ |
64 $$\mathfig{.9}{deligne/manifolds}$$ |
65 \caption{A fat graph}\label{delfig2}\end{figure} |
65 \caption{A fat graph}\label{delfig2}\end{figure} |
66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
68 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to |
68 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to |
69 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
69 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
80 Putting this together we get |
80 Putting this together we get |
81 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
81 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
82 \label{prop:deligne} |
82 \label{prop:deligne} |
83 There is a collection of maps |
83 There is a collection of maps |
84 \begin{eqnarray*} |
84 \begin{eqnarray*} |
85 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes |
85 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
86 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
86 \mapinf(\bc_*(M_{k}), \bc_*(N_{k})) & \\ |
87 & \hspace{-11em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
87 & \hspace{-11em}\to \mapinf(\bc_*(M_0), \bc_*(N_0)) |
88 \end{eqnarray*} |
88 \end{eqnarray*} |
89 which satisfy an operad type compatibility condition. \nn{spell this out} |
89 which satisfy an operad type compatibility condition. \nn{spell this out} |
90 \end{prop} |
90 \end{prop} |
91 |
91 |
92 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |
92 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |