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1 %!TEX root = ../blob1.tex |
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2 |
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 |
5 In this section we prove a higher dimensional version of the Deligne conjecture |
6 sketch |
6 about the action of the little disks operad on Hochschild cochains. |
7 \nn{revisit ``sketch" after proof is done} |
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8 the proof of a higher dimensional version of the Deligne conjecture |
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9 about the action of the little disks operad on Hochschild cohomology. |
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10 The first several paragraphs lead up to a precise statement of the result |
7 The first several paragraphs lead up to a precise statement of the result |
11 (Theorem \ref{thm:deligne} below). |
8 (Theorem \ref{thm:deligne} below). |
12 Then we sketch the proof. |
9 Then we give the proof. |
13 |
10 |
14 \nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, |
11 \nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, |
15 that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
12 that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
16 |
13 |
17 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
14 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
21 The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} gives a map |
18 The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} gives a map |
22 \[ |
19 \[ |
23 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
20 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
24 \to Hoch^*(C, C) . |
21 \to Hoch^*(C, C) . |
25 \] |
22 \] |
26 Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
23 Here $LD_k$ is the $k$-th space of the little disks operad and $Hoch^*(C, C)$ denotes Hochschild |
27 cochains. |
24 cochains. |
28 The little disks operad is homotopy equivalent to the |
25 The little disks operad is homotopy equivalent to the |
29 (transversely orient) fat graph operad |
26 (transversely oriented) fat graph operad |
30 \nn{need ref, or say more precisely what we mean}, |
27 (see below), |
31 and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
28 and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
32 of the blob complex of the interval, thought of as a bimodule for itself. |
29 of the blob complex of the interval, thought of as a bimodule for itself. |
33 \nn{need to make sure we prove this above}. |
30 (see \S\ref{ss:module-morphisms}). |
34 So the 1-dimensional Deligne conjecture can be restated as |
31 So the 1-dimensional Deligne conjecture can be restated as |
35 \[ |
32 \[ |
36 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
33 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
37 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
34 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
38 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
35 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
71 \caption{An $n$-dimensional fat graph}\label{delfig2} |
68 \caption{An $n$-dimensional fat graph}\label{delfig2} |
72 \end{figure} |
69 \end{figure} |
73 |
70 |
74 More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
71 More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
75 \begin{itemize} |
72 \begin{itemize} |
76 \item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
73 \item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-manifolds $N_0,\ldots,N_k$, |
77 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
74 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
78 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
75 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
79 boundaries. |
76 boundaries. |
80 \item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$. |
77 \item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$. |
81 %(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
78 %(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
108 leaving the $M_i$ and $N_i$ fixed. |
105 leaving the $M_i$ and $N_i$ fixed. |
109 (Keep in mind the case $R'_i = R_i$.) |
106 (Keep in mind the case $R'_i = R_i$.) |
110 (See Figure \ref{xdfig3}.) |
107 (See Figure \ref{xdfig3}.) |
111 \begin{figure}[t] |
108 \begin{figure}[t] |
112 $$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
109 $$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
113 \caption{Conjugating by a homeomorphism}\label{xdfig3} |
110 \caption{Conjugating by a homeomorphism |
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111 \nn{change right $R_i$ to $R'_i$}}\label{xdfig3} |
114 \end{figure} |
112 \end{figure} |
115 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
113 \item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
116 compatible disjoint union of $\bd M = \bd N$), we can replace |
114 compatible disjoint union of $\bd M = \bd N$), we can replace |
117 \begin{eqnarray*} |
115 \begin{eqnarray*} |
118 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
116 (\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
168 Let $f_i = \rm{id}$ for all $i$. |
166 Let $f_i = \rm{id}$ for all $i$. |
169 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
167 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
170 with contractible fibers. |
168 with contractible fibers. |
171 (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
169 (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
172 direction without changing their ordering.) |
170 direction without changing their ordering.) |
173 \nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
171 %\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
174 does this need more explanation?} |
172 %does this need more explanation?} |
175 |
173 |
176 Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to |
174 Another familiar subspace of the $n$-FG operad is $\Homeo(M_0\to N_0)$, which corresponds to |
177 case $k=0$ (no holes). |
175 case $k=0$ (no holes). |
178 |
176 |
179 \medskip |
177 \medskip |
180 |
178 |
181 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
179 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
192 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
190 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
193 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
191 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
194 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
192 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
195 \stackrel{f_k}{\to} \bc_*(N_0) |
193 \stackrel{f_k}{\to} \bc_*(N_0) |
196 \] |
194 \] |
197 (Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
195 (Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}s.) |
198 \nn{need to double check case where $\alpha_i$'s are not closed.} |
196 \nn{need to double check case where $\alpha_i$'s are not closed.} |
199 It is easy to check that the above definition is compatible with the equivalence relations |
197 It is easy to check that the above definition is compatible with the equivalence relations |
200 and also the operad structure. |
198 and also the operad structure. |
201 We can reinterpret the above as a chain map |
199 We can reinterpret the above as a chain map |
202 \[ |
200 \[ |
235 \] |
233 \] |
236 It suffices to show that the above maps are compatible with the relations whereby |
234 It suffices to show that the above maps are compatible with the relations whereby |
237 $FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
235 $FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
238 This in turn follows easily from the fact that |
236 This in turn follows easily from the fact that |
239 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
237 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
240 |
238 %\nn{should add some detail to above} |
241 \nn{should add some detail to above} |
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242 \end{proof} |
239 \end{proof} |
243 |
240 |
244 \nn{maybe point out that even for $n=1$ there's something new here.} |
241 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
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242 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |
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243 in a different pattern from $M_i$. |
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244 The genus of the fat graph could be greater than zero. |