4 \label{sec:deligne} |
4 \label{sec:deligne} |
5 In this section we discuss Property \ref{property:deligne}, |
5 In this section we discuss Property \ref{property:deligne}, |
6 \begin{prop}[Higher dimensional Deligne conjecture] |
6 \begin{prop}[Higher dimensional Deligne conjecture] |
7 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
7 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
8 \end{prop} |
8 \end{prop} |
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9 |
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10 We will give a more precise statement of the proposition below. |
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11 |
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12 \nn{for now, we just sketch the proof.} |
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13 |
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14 \def\mapinf{\Maps_\infty} |
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15 |
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16 The usual Deligne conjecture \nn{need refs} gives a map |
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17 \[ |
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18 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
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19 \to Hoch^*(C, C) . |
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20 \] |
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21 Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
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22 cochains. |
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23 The little disks operad is homotopy equivalent to the fat graph operad |
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24 \nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
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25 of the blob complex of the interval. |
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26 \nn{need to make sure we prove this above}. |
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27 So the 1-dimensional Deligne conjecture can be restated as |
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28 \begin{eqnarray*} |
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29 C_*(FG_k)\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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30 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\ |
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31 & \hspace{-5em} \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
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32 \end{eqnarray*} |
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33 See Figure \ref{delfig1}. |
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34 \begin{figure}[!ht] |
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35 $$\mathfig{.9}{tempkw/delfig1}$$ |
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36 \caption{A fat graph}\label{delfig1}\end{figure} |
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37 |
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38 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
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39 of Figure \ref{delfig1} and ending at the topmost interval. |
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40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
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41 We remove the bottom interval of the bigon and replace it with the top interval. |
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42 To map this topological operation to an algebraic one, we need, for each hole, element of |
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43 $\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
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44 So for each fixed fat graph we have a map |
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45 \[ |
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46 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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47 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
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48 \] |
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49 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
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50 between the maps associated to the endpoints of the 1-chain. |
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51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
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52 |
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53 It should now be clear how to generalize this to higher dimensions. |
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54 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
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55 involved were 1-dimensional. |
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56 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries |
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57 on an $n$-manifold. |
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58 More specifically, \nn{...} |
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59 |
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60 |
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61 \medskip |
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62 \hrule\medskip |
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63 |
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64 |
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65 Figure \ref{delfig2} |
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66 \begin{figure}[!ht] |
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67 $$\mathfig{.9}{tempkw/delfig2}$$ |
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68 \caption{A fat graph}\label{delfig2}\end{figure} |
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69 |
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70 |
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71 \begin{eqnarray*} |
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72 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes |
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73 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
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74 & \hspace{-5em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
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75 \end{eqnarray*} |
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76 |
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77 \medskip |
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78 \hrule\medskip |
9 |
79 |
10 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
80 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
11 of $n$-manifolds |
81 of $n$-manifolds |
12 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
82 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
13 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
83 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |