35 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
35 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
36 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
36 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
37 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
37 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
38 \] |
38 \] |
39 See Figure \ref{delfig1}. |
39 See Figure \ref{delfig1}. |
40 \begin{figure}[!ht] |
40 \begin{figure}[t] |
41 $$\mathfig{.9}{deligne/intervals}$$ |
41 $$\mathfig{.9}{deligne/intervals}$$ |
42 \caption{A fat graph}\label{delfig1}\end{figure} |
42 \caption{A fat graph}\label{delfig1}\end{figure} |
43 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
43 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
44 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
44 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
45 morphisms of such modules as defined in |
45 morphisms of such modules as defined in |
63 It should now be clear how to generalize this to higher dimensions. |
63 It should now be clear how to generalize this to higher dimensions. |
64 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
64 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
65 involved were 1-dimensional. |
65 involved were 1-dimensional. |
66 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
66 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
67 on an $n$-manifold (Figure \ref{delfig2}). |
67 on an $n$-manifold (Figure \ref{delfig2}). |
68 \begin{figure}[!ht] |
68 \begin{figure}[t] |
69 $$\mathfig{.9}{deligne/manifolds}$$ |
69 $$\mathfig{.9}{deligne/manifolds}$$ |
70 \caption{An $n$-dimensional fat graph}\label{delfig2} |
70 \caption{An $n$-dimensional fat graph}\label{delfig2} |
71 \end{figure} |
71 \end{figure} |
72 |
72 |
73 More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
73 More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
74 \begin{itemize} |
74 \begin{itemize} |
75 \item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
75 \item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
86 \end{eqnarray*} |
86 \end{eqnarray*} |
87 Each $f_i$ should be the identity restricted to $E_0$. |
87 Each $f_i$ should be the identity restricted to $E_0$. |
88 \end{itemize} |
88 \end{itemize} |
89 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
89 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
90 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
90 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
91 (see Figure xxxx). |
91 (see Figure \ref{xdfig2}). |
|
92 \begin{figure}[t] |
|
93 $$\mathfig{.9}{tempkw/dfig2}$$ |
|
94 \caption{$n$-dimensional fat graph from mapping cylinders}\label{xdfig2} |
|
95 \end{figure} |
92 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
96 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
93 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
97 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
94 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
98 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
95 cylinders. |
99 cylinders. |
96 More specifically, we impose the following two equivalence relations: |
100 More specifically, we impose the following two equivalence relations: |
100 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
104 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
101 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
105 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
102 \end{eqnarray*} |
106 \end{eqnarray*} |
103 leaving the $M_i$ and $N_i$ fixed. |
107 leaving the $M_i$ and $N_i$ fixed. |
104 (Keep in mind the case $R'_i = R_i$.) |
108 (Keep in mind the case $R'_i = R_i$.) |
105 (See Figure xxxx.) |
109 (See Figure \ref{xdfig3}.) |
|
110 \begin{figure}[t] |
|
111 $$\mathfig{.9}{tempkw/dfig3}$$ |
|
112 \caption{Conjugating by a homeomorphism}\label{xdfig3} |
|
113 \end{figure} |
106 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
114 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
107 compatible disjoint union of $\bd M = \bd N$), we can replace |
115 compatible disjoint union of $\bd M = \bd N$), we can replace |
108 \begin{eqnarray*} |
116 \begin{eqnarray*} |
109 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
117 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
110 (\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
118 (\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
111 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& |
119 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& |
112 (\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\ |
120 (\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\ |
113 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
121 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
114 \end{eqnarray*} |
122 \end{eqnarray*} |
115 (See Figure xxxx.) |
123 (See Figure \ref{xdfig1}.) |
|
124 \begin{figure}[t] |
|
125 $$\mathfig{.9}{tempkw/dfig1}$$ |
|
126 \caption{Changing the order of a surgery}\label{xdfig1} |
|
127 \end{figure} |
116 \end{itemize} |
128 \end{itemize} |
117 |
129 |
118 Note that the second equivalence increases the number of holes (or arity) by 1. |
130 Note that the second equivalence increases the number of holes (or arity) by 1. |
119 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
131 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
120 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
132 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |