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3 \section{Higher-dimensional Deligne conjecture} |
3 \section{Higher-dimensional Deligne conjecture} |
4 \label{sec:deligne} |
4 \label{sec:deligne} |
5 In this section we discuss Property \ref{property:deligne}, |
5 In this section we discuss |
6 \begin{prop}[Higher dimensional Deligne conjecture] |
6 \newenvironment{property:deligne}{\textbf{Property \ref{property:deligne} (Higher dimensional Deligne conjecture)}\it}{} |
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8 \begin{property:deligne} |
7 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
9 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
8 \end{prop} |
10 \end{property:deligne} |
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10 We will give a more precise statement of the proposition below. |
12 We will state this more precisely below as Proposition \ref{prop:deligne}, and just sketch a proof. First, we recall the usual Deligne conjecture, explain how to think of it as a statement about blob complexes, and begin to generalize it. |
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12 \nn{for now, we just sketch the proof.} |
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14 \def\mapinf{\Maps_\infty} |
14 \def\mapinf{\Maps_\infty} |
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16 The usual Deligne conjecture \nn{need refs} gives a map |
16 The usual Deligne conjecture \nn{need refs} gives a map |
17 \[ |
17 \[ |
75 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
75 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
76 The ``holes" in the above |
76 The ``holes" in the above |
77 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
77 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
78 \nn{need to make up my mind which notation I'm using for the module maps} |
78 \nn{need to make up my mind which notation I'm using for the module maps} |
79 |
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80 Putting this together we get a collection of maps |
80 Putting this together we get |
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81 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
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82 \label{prop:deligne} |
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83 There is a collection of maps |
81 \begin{eqnarray*} |
84 \begin{eqnarray*} |
82 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_0), \bc_*(N_0))\otimes\cdots\otimes |
83 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
86 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
84 & \hspace{-11em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
87 & \hspace{-11em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
85 \end{eqnarray*} |
88 \end{eqnarray*} |
86 which satisfy an operad type compatibility condition. |
89 which satisfy an operad type compatibility condition. \nn{spell this out} |
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90 \end{prop} |
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88 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}. |
89 And indeed, the proof is very similar \nn{...} |
93 And indeed, the proof is very similar \nn{...} |
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95 |