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57 It should now be clear how to generalize this to higher dimensions. |
57 It should now be clear how to generalize this to higher dimensions. |
58 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
58 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
59 involved were 1-dimensional. |
59 involved were 1-dimensional. |
60 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
60 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
61 on an $n$-manifold. |
61 on an $n$-manifold (Figure \ref{delfig2}). |
62 |
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63 \nn{*** resume revising here} |
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64 |
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65 More specifically, |
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66 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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67 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
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68 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
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69 (See Figure \ref{delfig2}.) |
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70 \begin{figure}[!ht] |
62 \begin{figure}[!ht] |
71 $$\mathfig{.9}{deligne/manifolds}$$ |
63 $$\mathfig{.9}{deligne/manifolds}$$ |
72 \caption{A fat graph}\label{delfig2} |
64 \caption{An $n$-dimensional fat graph}\label{delfig2} |
73 \end{figure} |
65 \end{figure} |
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66 |
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67 More specifically, an $n$-dimensional fat graph consists of: |
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68 \begin{itemize} |
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69 \item ``Incoming" $n$-manifolds $M_1,\ldots,M_k$ and ``outgoing" $n$-manifolds $N_1,\ldots,N_k$, |
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70 with $\bd M_i = \bd N_i$ for all $i$. |
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71 \item An ``outer boundary" $n{-}1$-manifold $E$. |
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72 \item Additional manifolds $R_0,\ldots,R_{k+1}$, with $\bd R_i = E\cup \bd M_i = E\cup \bd N_i$. |
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73 (By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
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74 We call $R_0$ the outer incoming manifold and $R_{k+1}$ the outer outgoing manifold |
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75 \item Homeomorphisms $f_i : R_i\cup N_i\to R_{i+1}\cup M_{i+1}$, $0\le i \le k$. |
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76 \end{itemize} |
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77 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
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78 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$. |
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79 \nn{need figure} |
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83 \nn{*** resume revising here} |
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80 The components of the $n$-dimensional fat graph operad are indexed by tuples |
87 The components of the $n$-dimensional fat graph operad are indexed by tuples |
81 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
88 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |