6 about the action of the little disks operad on Hochschild cochains. |
6 about the action of the little disks operad on Hochschild cochains. |
7 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 |
8 (Theorem \ref{thm:deligne} below). |
8 (Theorem \ref{thm:deligne} below). |
9 Then we give the proof. |
9 Then we give the proof. |
10 |
10 |
11 %\nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, |
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12 %that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
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13 |
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14 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
11 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
15 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
12 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
16 |
13 |
17 |
14 |
18 The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} gives a map |
15 The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} gives a map |
20 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
17 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
21 \to Hoch^*(C, C) . |
18 \to Hoch^*(C, C) . |
22 \] |
19 \] |
23 Here $LD_k$ is the $k$-th space of the little disks operad and $Hoch^*(C, C)$ denotes Hochschild |
20 Here $LD_k$ is the $k$-th space of the little disks operad and $Hoch^*(C, C)$ denotes Hochschild |
24 cochains. |
21 cochains. |
25 The little disks operad is homotopy equivalent to the |
22 |
26 (transversely oriented) fat graph operad |
23 We now reinterpret $C_*(LD_k)$ and $Hoch^*(C, C)$ in such a way as to make the generalization to |
27 (see below), |
24 higher dimensions clear. |
28 and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
25 |
29 of the blob complex of the interval, thought of as a bimodule for itself. |
26 The little disks operad is homotopy equivalent to configurations of little bigons inside a big bigon, |
30 (see \S\ref{ss:module-morphisms}). |
27 as shown in Figure \ref{delfig1}. |
31 So the 1-dimensional Deligne conjecture can be restated as |
28 We can think of such a configuration as encoding a sequence of surgeries, starting at the bottommost interval |
32 \[ |
29 of Figure \ref{delfig1} and ending at the topmost interval. |
33 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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34 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
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35 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
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36 \] |
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37 See Figure \ref{delfig1}. |
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38 \begin{figure}[t] |
30 \begin{figure}[t] |
39 $$\mathfig{.9}{deligne/intervals}$$ |
31 $$\mathfig{.9}{deligne/intervals}$$ |
40 \caption{A fat graph}\label{delfig1}\end{figure} |
32 \caption{Little bigons, though of as encoding surgeries}\label{delfig1}\end{figure} |
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33 The surgeries correspond to the $k$ bigon-shaped ``holes". |
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34 We remove the bottom interval of each little bigon and replace it with the top interval. |
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35 To convert this topological operation to an algebraic one, we need, for each hole, an element of |
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36 $\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$, which is homotopy equivalent to $Hoch^*(C, C)$. |
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37 So for each fixed configuration we have a map |
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38 \[ |
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39 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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40 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
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41 \] |
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42 If we deform the configuration, corresponding to a 1-chain in $C_*(LD_k)$, we get a homotopy |
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43 between the maps associated to the endpoints of the 1-chain. |
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44 Similarly, higher-dimensional chains in $C_*(LD_k)$ give rise to higher homotopies. |
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45 |
41 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
46 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
42 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
47 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
43 morphisms of such modules as defined in |
48 morphisms of such modules as defined in |
44 \S\ref{ss:module-morphisms}. |
49 \S\ref{ss:module-morphisms}. |
45 |
50 |
46 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
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47 of Figure \ref{delfig1} and ending at the topmost interval. |
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48 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
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49 We remove the bottom interval of the bigon and replace it with the top interval. |
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50 To convert this topological operation to an algebraic one, we need, for each hole, an element of |
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51 $\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
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52 So for each fixed fat graph we have a map |
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53 \[ |
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54 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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55 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
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56 \] |
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57 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
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58 between the maps associated to the endpoints of the 1-chain. |
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59 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
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60 |
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61 It should now be clear how to generalize this to higher dimensions. |
51 It should now be clear how to generalize this to higher dimensions. |
62 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
52 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
63 involved were 1-dimensional. |
53 involved were 1-dimensional. |
64 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
54 So we will define, below, the operad of $n$-dimensional surgery cylinders, analogous to mapping |
65 on an $n$-manifold (Figure \ref{delfig2}). |
55 cylinders of homeomorphisms (Figure \ref{delfig2}). |
66 \begin{figure}[t] |
56 \begin{figure}[t] |
67 $$\mathfig{.9}{deligne/manifolds}$$ |
57 $$\mathfig{.9}{deligne/manifolds}$$ |
68 \caption{An $n$-dimensional fat graph}\label{delfig2} |
58 \caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
69 \end{figure} |
59 \end{figure} |
70 |
60 (Note that $n$ is the dimension of the manifolds we are doing surgery on; the surgery cylinders |
71 More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: |
61 are $n{+}1$-dimensional.) |
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62 |
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63 An $n$-dimensional surgery cylinder ($n$-SC for short) consists of: |
72 \begin{itemize} |
64 \begin{itemize} |
73 \item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-manifolds $N_0,\ldots,N_k$, |
65 \item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-manifolds $N_0,\ldots,N_k$, |
74 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
66 with $\bd M_i = \bd N_i = E_i$ for all $i$. |
75 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
67 We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
76 boundaries. |
68 boundaries. |
87 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
79 We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
88 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
80 with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
89 (see Figure \ref{xdfig2}). |
81 (see Figure \ref{xdfig2}). |
90 \begin{figure}[t] |
82 \begin{figure}[t] |
91 $$\mathfig{.9}{deligne/mapping-cylinders}$$ |
83 $$\mathfig{.9}{deligne/mapping-cylinders}$$ |
92 \caption{An $n$-dimensional fat graph constructed from mapping cylinders}\label{xdfig2} |
84 \caption{An $n$-dimensional surgery cylinder constructed from mapping cylinders}\label{xdfig2} |
93 \end{figure} |
85 \end{figure} |
94 The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
86 %The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
95 We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
87 We regard two such surgery cylinders as the same if there is a homeomorphism between them which is the |
96 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
88 identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
97 cylinders. |
89 cylinders. |
98 More specifically, we impose the following two equivalence relations: |
90 More specifically, we impose the following two equivalence relations: |
99 \begin{itemize} |
91 \begin{itemize} |
100 \item If $g: R_i\to R'_i$ is a homeomorphism, we can replace |
92 \item If $g: R_i\to R'_i$ is a homeomorphism, we can replace |
129 Note that the second equivalence increases the number of holes (or arity) by 1. |
121 Note that the second equivalence increases the number of holes (or arity) by 1. |
130 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
122 We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed. |
131 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
123 In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
132 do not overlap, we can perform them in reverse order or simultaneously. |
124 do not overlap, we can perform them in reverse order or simultaneously. |
133 |
125 |
134 There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
126 There is an operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary |
135 of one graph into one of the inner boundaries of another graph. |
127 of one cylinder into one of the inner boundaries of another cylinder. |
136 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
128 We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
137 |
129 |
138 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
130 For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
139 $FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
131 $SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above. |
140 (Note that in different parts of $FG^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s |
132 (Note that in different parts of $SC^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s |
141 are ordered differently.) |
133 are ordered differently.) |
142 The topology comes from the spaces |
134 The topology comes from the spaces |
143 \[ |
135 \[ |
144 \Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
136 \Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
145 \cdots\times \Homeo(R_k\cup N_k\to N_0) |
137 \cdots\times \Homeo(R_k\cup N_k\to N_0) |
146 \] |
138 \] |
147 and the above equivalence relations. |
139 and the above equivalence relations. |
148 We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
140 We will denote the typical element of $SC^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
149 |
141 |
150 \medskip |
142 \medskip |
151 |
143 |
152 %The little $n{+}1$-ball operad injects into the $n$-FG operad. |
144 %The little $n{+}1$-balls operad injects into the $n$-SC operad. |
153 The $n$-FG operad contains the little $n{+}1$-balls operad. |
145 The $n$-SC operad contains the little $n{+}1$-balls operad. |
154 Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
146 Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
155 $n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
147 $n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
156 and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
148 and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
157 More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. |
149 More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. |
158 Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with |
150 Let $z$ be a point of the $k$-th space of the little $n{+}1$-balls operad, with |
159 little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
151 little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
160 We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. |
152 We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. |
161 Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. |
153 Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. |
162 Let $B\sub\r^n$ be the standard $n$-ball. |
154 Let $B\sub\r^n$ be the standard $n$-ball. |
163 Let $M_i$ and $N_i$ be $B$ for all $i$. |
155 Let $M_i$ and $N_i$ be $B$ for all $i$. |
164 Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
156 Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
165 Let $R_i = B\setmin \pi(D_i)$. |
157 Let $R_i = B\setmin \pi(D_i)$. |
166 Let $f_i = \rm{id}$ for all $i$. |
158 Let $f_i = \rm{id}$ for all $i$. |
167 We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, |
159 We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad, |
168 with contractible fibers. |
160 with contractible fibers. |
169 (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
161 (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
170 direction without changing their ordering.) |
162 direction without changing their ordering.) |
171 %\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
163 %\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
172 %does this need more explanation?} |
164 %does this need more explanation?} |
173 |
165 |
174 Another familiar subspace of the $n$-FG operad is $\Homeo(M_0\to N_0)$, which corresponds to |
166 Another familiar subspace of the $n$-SC operad is $\Homeo(M_0\to N_0)$, which corresponds to |
175 case $k=0$ (no holes). |
167 case $k=0$ (no holes). |
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168 In this case the surgery cylinder is just a single mapping cylinder. |
176 |
169 |
177 \medskip |
170 \medskip |
178 |
171 |
179 Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
172 Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$. |
180 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
173 Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
181 as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
174 as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
182 We define a map |
175 We define a map |
183 \[ |
176 \[ |
184 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
177 p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
195 (Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}.) |
188 (Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}.) |
196 It is easy to check that the above definition is compatible with the equivalence relations |
189 It is easy to check that the above definition is compatible with the equivalence relations |
197 and also the operad structure. |
190 and also the operad structure. |
198 We can reinterpret the above as a chain map |
191 We can reinterpret the above as a chain map |
199 \[ |
192 \[ |
200 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
193 p: C_0(SC^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
201 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
194 \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
202 \] |
195 \] |
203 The main result of this section is that this chain map extends to the full singular |
196 The main result of this section is that this chain map extends to the full singular |
204 chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. |
197 chain complex $C_*(SC^n_{\ol{M}\ol{N}})$. |
205 |
198 |
206 \begin{thm} |
199 \begin{thm} |
207 \label{thm:deligne} |
200 \label{thm:deligne} |
208 There is a collection of chain maps |
201 There is a collection of chain maps |
209 \[ |
202 \[ |
210 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
203 C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
211 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
204 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
212 \] |
205 \] |
213 which satisfy the operad compatibility conditions. |
206 which satisfy the operad compatibility conditions. |
214 On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
207 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
215 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. |
208 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. |
216 \end{thm} |
209 \end{thm} |
217 |
210 |
218 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
211 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
219 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on |
212 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on |
220 blob cochains. |
213 blob cochains. |
221 As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes |
214 As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
222 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. |
215 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |
223 |
216 |
224 \begin{proof} |
217 \begin{proof} |
225 As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
218 As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint |
226 union of products of homeomorphism spaces, modulo some relations. |
219 union of products of homeomorphism spaces, modulo some relations. |
227 By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
220 By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
228 a chain map |
221 a chain map |
229 \[ |
222 \[ |
230 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
223 C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
231 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
224 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
232 \] |
225 \] |
233 It suffices to show that the above maps are compatible with the relations whereby |
226 It suffices to show that the above maps are compatible with the relations whereby |
234 $FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
227 $SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. |
235 This in turn follows easily from the fact that |
228 This in turn follows easily from the fact that |
236 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
229 the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. |
237 %\nn{should add some detail to above} |
230 %\nn{should add some detail to above} |
238 \end{proof} |
231 \end{proof} |
239 |
232 |
240 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
233 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
241 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |
234 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |
242 in a different pattern from $M_i$. |
235 in a different pattern from $M_i$. |
243 The genus of the fat graph could be greater than zero. |
236 The genus of the surface associated to the surgery cylinder could be greater than zero. |