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 consists of: |
73 More specifically, an $n$-dimensional fat graph consists of: |
74 \begin{itemize} |
74 \begin{itemize} |
75 \item ``Incoming" $n$-manifolds $M_1,\ldots,M_k$ and ``outgoing" $n$-manifolds $N_1,\ldots,N_k$, |
75 \item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
76 with $\bd M_i = \bd N_i$ for all $i$. |
76 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
77 \item An ``outer boundary" $n{-}1$-manifold $E$. |
77 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
78 \item Additional manifolds $R_0,\ldots,R_{k+1}$, with $\bd R_i = E\cup \bd M_i = E\cup \bd N_i$. |
78 boundaries. |
79 (By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
79 \item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$. |
80 We call $R_0$ the outer incoming manifold and $R_{k+1}$ the outer outgoing manifold |
80 %(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
81 \item Homeomorphisms $f_i : R_i\cup N_i\to R_{i+1}\cup M_{i+1}$, $0\le i \le k$. |
81 \item Homeomorphisms |
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82 \begin{eqnarray*} |
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83 f_0: M_0 &\to& R_1\cup M_1 \\ |
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84 f_i: R_i\cup N_i &\to& R_{i+1}\cup M_{i+1}\;\; \mbox{for}\, 1\le i \le k-1 \\ |
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85 f_k: R_k\cup N_k &\to& N_0 . |
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86 \end{eqnarray*} |
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87 Each $f_i$ should be the identity restricted to $E_0$. |
82 \end{itemize} |
88 \end{itemize} |
83 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 $C(f_0),\ldots,C(f_k)$, |
84 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 $R_{i+1}$ |
85 (see Figure xxxx). |
91 (see Figure xxxx). |
86 \nn{also need to revise outer labels of older fig} |
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87 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
92 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
88 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
93 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
89 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
94 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
90 cylinders. |
95 cylinders. |
91 More specifically, we impose the following two equivalence relations: |
96 More specifically, we impose the following two equivalence relations: |
92 \begin{itemize} |
97 \begin{itemize} |
93 \item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
98 \item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
94 \[ |
99 \[ |
95 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots) . |
100 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots) . |
96 \] |
101 \] |
97 (See Figure xxx.) |
102 (See Figure xxxx.) |
98 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
103 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
99 compatible disjoint union of $\bd M = \bd N$), we can replace |
104 compatible disjoint union of $\bd M = \bd N$), we can replace |
100 \begin{eqnarray*} |
105 \begin{eqnarray*} |
101 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
106 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
102 (\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
107 (\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
110 Note that the second equivalence increases the number of holes (or arity) by 1. |
115 Note that the second equivalence increases the number of holes (or arity) by 1. |
111 We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed. |
116 We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed. |
112 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
117 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
113 do not overlap, we can perform them in reverse order or simultaneously. |
118 do not overlap, we can perform them in reverse order or simultaneously. |
114 |
119 |
115 \nn{operad structure (need to ntro mroe terminology above} |
120 There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
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121 of one graph into one of the inner boundaries of another graph. |
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122 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
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123 |
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124 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
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125 $FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
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126 The topology comes from the spaces |
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127 \[ |
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128 \Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
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129 \cdots\times \Homeo(R_k\cup N_k\to N_0) |
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130 \] |
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131 and the above equivalence relations. |
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132 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
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133 |
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134 |
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135 \medskip |
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136 |
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137 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
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138 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
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139 as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
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140 We define a map |
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141 \[ |
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142 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
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143 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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144 \] |
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145 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$ to be the composition |
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146 \[ |
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147 \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
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148 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
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149 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \to |
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150 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
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151 \stackrel{f_k}{\to} \bc_*(N_0) |
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152 \] |
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153 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
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154 It is easy to check that the above definition is compatible with the equivalence relations |
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155 and also the operad structure. |
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156 |
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157 \nn{little m-disks operad; } |
116 |
158 |
117 \nn{*** resume revising here} |
159 \nn{*** resume revising here} |
118 |
160 |
119 |
161 |
120 |
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121 The components of the $n$-dimensional fat graph operad are indexed by tuples |
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122 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
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123 \nn{not quite true: this is coarser than components} |
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124 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to |
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125 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
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126 \nn{what about rotating in the horizontal directions?} |
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127 |
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128 |
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129 If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define |
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130 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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131 $A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both |
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132 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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133 The ``holes" in the above |
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134 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
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135 \nn{need to make up my mind which notation I'm using for the module maps} |
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136 |
162 |
137 Putting this together we get |
163 Putting this together we get |
138 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
164 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
139 \label{prop:deligne} |
165 \label{prop:deligne} |
140 There is a collection of maps |
166 There is a collection of maps |