68 \begin{figure}[!ht] |
68 \begin{figure}[!ht] |
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 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 $N_0,\ldots,N_k$, |
76 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
76 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
77 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
77 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
78 boundaries. |
78 boundaries. |
93 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 |
94 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 |
95 cylinders. |
95 cylinders. |
96 More specifically, we impose the following two equivalence relations: |
96 More specifically, we impose the following two equivalence relations: |
97 \begin{itemize} |
97 \begin{itemize} |
98 \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 |
99 \[ |
99 \begin{eqnarray*} |
100 (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
100 (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
101 \] |
101 (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
102 leaving the $M_i$, $N_i$ and $R_i$ fixed. |
102 \end{eqnarray*} |
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103 leaving the $M_i$ and $N_i$ fixed. |
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104 (Keep in mind the case $R'_i = R_i$.) |
103 (See Figure xxxx.) |
105 (See Figure xxxx.) |
104 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
106 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
105 compatible disjoint union of $\bd M = \bd N$), we can replace |
107 compatible disjoint union of $\bd M = \bd N$), we can replace |
106 \begin{eqnarray*} |
108 \begin{eqnarray*} |
107 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
109 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
112 \end{eqnarray*} |
114 \end{eqnarray*} |
113 (See Figure xxxx.) |
115 (See Figure xxxx.) |
114 \end{itemize} |
116 \end{itemize} |
115 |
117 |
116 Note that the second equivalence increases the number of holes (or arity) by 1. |
118 Note that the second equivalence increases the number of holes (or arity) by 1. |
117 We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed. |
119 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
118 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
120 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
119 do not overlap, we can perform them in reverse order or simultaneously. |
121 do not overlap, we can perform them in reverse order or simultaneously. |
120 |
122 |
121 There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
123 There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
122 of one graph into one of the inner boundaries of another graph. |
124 of one graph into one of the inner boundaries of another graph. |
123 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
125 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
124 |
126 |
125 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
127 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
126 $FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
128 $FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
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129 (Note that in different parts of $FG^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s |
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130 are ordered differently.) |
127 The topology comes from the spaces |
131 The topology comes from the spaces |
128 \[ |
132 \[ |
129 \Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
133 \Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
130 \cdots\times \Homeo(R_k\cup N_k\to N_0) |
134 \cdots\times \Homeo(R_k\cup N_k\to N_0) |
131 \] |
135 \] |
132 and the above equivalence relations. |
136 and the above equivalence relations. |
133 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
137 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
134 |
138 |
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139 \medskip |
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140 |
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141 %The little $n{+}1$-ball operad injects into the $n$-FG operad. |
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142 The $n$-FG operad contains the little $n{+}1$-ball operad. |
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143 Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
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144 $n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
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145 and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
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146 More precisely, let $x_0,\ldots,x_n$ be the coordinates of $\r^{n+1}$. |
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147 Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with |
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148 little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
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149 We assume the $D_i$'s are ordered according to the $x_n$ coordinate of their centers. |
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150 Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_n$. |
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151 Let $B\sub\r^n$ be the standard $n$-ball. |
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152 Let $M_i$ and $N_i$ be $B$ for all $i$. |
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153 Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
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154 Let $R_i = B\setmin \pi(D_i)$. |
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155 Let $f_i = \rm{id}$ for all $i$. |
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156 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
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157 with contractible fibers. |
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158 (The fibers correspond to moving the $D_i$'s in the $x_n$ direction without changing their ordering.) |
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159 \nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
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160 does this need more explanation?} |
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161 |
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162 Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to |
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163 case $k=0$ (no holes). |
135 |
164 |
136 \medskip |
165 \medskip |
137 |
166 |
138 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
167 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
139 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
168 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
141 We define a map |
170 We define a map |
142 \[ |
171 \[ |
143 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
172 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
144 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
173 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
145 \] |
174 \] |
146 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$ to be the composition |
175 Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition |
147 \[ |
176 \[ |
148 \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
177 \bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
149 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
178 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
150 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \to |
179 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
151 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
180 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
152 \stackrel{f_k}{\to} \bc_*(N_0) |
181 \stackrel{f_k}{\to} \bc_*(N_0) |
153 \] |
182 \] |
154 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
183 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
155 It is easy to check that the above definition is compatible with the equivalence relations |
184 It is easy to check that the above definition is compatible with the equivalence relations |
156 and also the operad structure. |
185 and also the operad structure. |
157 |
186 We can reinterpret the above as a chain map |
158 \nn{little m-disks operad; } |
187 \[ |
159 |
188 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
160 \nn{*** resume revising here} |
189 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
161 |
190 \] |
162 |
191 The main result of this section is that this chain map extends to the full singular |
163 |
192 chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. |
164 Putting this together we get |
193 |
165 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
194 \begin{prop} |
166 \label{prop:deligne} |
195 \label{prop:deligne} |
167 There is a collection of maps |
196 There is a collection of chain maps |
168 \begin{eqnarray*} |
197 \[ |
169 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
198 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
170 \hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\ |
199 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
171 & \hspace{-11em}\to \hom(\bc_*(M_0), \bc_*(N_0)) |
200 \] |
172 \end{eqnarray*} |
201 which satisfy the operad compatibility conditions. |
173 which satisfy an operad type compatibility condition. \nn{spell this out} |
202 On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
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203 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}. |
174 \end{prop} |
204 \end{prop} |
175 |
205 |
176 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |
206 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
177 And indeed, the proof is very similar \nn{...} |
207 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on |
178 |
208 blob cochains. |
179 |
209 As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes |
180 |
210 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. |
181 \medskip |
211 |
182 \hrule\medskip |
212 \nn{...} |
183 |
213 |
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214 \nn{maybe point out that even for $n=1$ there's something new here.} |