9 

    10 We will give a more precise statement of the proposition below.

    11 

    12 \nn{for now, we just sketch the proof.}

    13 

    14 \def\mapinf{\Maps_\infty}

    15 

    16 The usual Deligne conjecture \nn{need refs} gives a map

    17 \[

    18 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}

    19 			\to  Hoch^*(C, C) .

    20 \]

    21 Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild

    22 cochains.

    23 The little disks operad is homotopy equivalent to the fat graph operad

    24 \nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms

    25 of the blob complex of the interval.

    26 \nn{need to make sure we prove this above}.

    27 So the 1-dimensional Deligne conjecture can be restated as

    28 \begin{eqnarray*}

    29 	C_*(FG_k)\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots

    30 	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\

    31 	  & \hspace{-5em} \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .

    32 \end{eqnarray*}

    33 See Figure \ref{delfig1}.

    34 \begin{figure}[!ht]

    35 $$\mathfig{.9}{tempkw/delfig1}$$

    36 \caption{A fat graph}\label{delfig1}\end{figure}

    37 

    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.

    40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph.

    41 We remove the bottom interval of the bigon and replace it with the top interval.

    42 To map this topological operation to an algebraic one, we need, for each hole, element of

    43 $\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.

    44 So for each fixed fat graph we have a map

    45 \[

    46 	 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots

    47 	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))  \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .

    48 \]

    49 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy

    50 between the maps associated to the endpoints of the 1-chain.

    51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies.

    52 

    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

    55 involved were 1-dimensional.

    56 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries

    57 on an $n$-manifold.

    58 More specifically, \nn{...}

    59 

    60 

    61 \medskip

    62 \hrule\medskip

    63 

    64 

    65 Figure \ref{delfig2}

    66 \begin{figure}[!ht]

    67 $$\mathfig{.9}{tempkw/delfig2}$$

    68 \caption{A fat graph}\label{delfig2}\end{figure}

    69 

    70 

    71 \begin{eqnarray*}

    72 	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes 

    73 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\

    74 	& \hspace{-5em}\to  \mapinf(\bc_*(M_k), \bc_*(N_k))

    75 \end{eqnarray*}

    76 

    77 \medskip

    78 \hrule\medskip