1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
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 discuss |
5 In this section we |
6 \newenvironment{property:deligne}{\textbf{Property \ref{property:deligne} (Higher dimensional Deligne conjecture)}\it}{} |
6 sketch |
7 |
7 \nn{revisit ``sketch" after proof is done} |
8 \begin{property:deligne} |
8 the proof of a higher dimensional version of the Deligne conjecture |
9 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
9 about the action of the little disks operad on Hochschild cohomology. |
10 \end{property:deligne} |
10 The first several paragraphs lead up to a precise statement of the result |
11 |
11 (Proposition \ref{prop:deligne} below). |
12 We will state this more precisely below as Proposition \ref{prop:deligne}, and just sketch a proof. First, we recall the usual Deligne conjecture, explain how to think of it as a statement about blob complexes, and begin to generalize it. |
12 Then we sketch the proof. |
13 |
|
14 %\def\mapinf{\Maps_\infty} |
|
15 |
13 |
16 The usual Deligne conjecture \nn{need refs} gives a map |
14 The usual Deligne conjecture \nn{need refs} gives a map |
17 \[ |
15 \[ |
18 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
16 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
19 \to Hoch^*(C, C) . |
17 \to Hoch^*(C, C) . |
20 \] |
18 \] |
21 Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
19 Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
22 cochains. |
20 cochains. |
23 The little disks operad is homotopy equivalent to the fat graph operad |
21 The little disks operad is homotopy equivalent to the |
24 \nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
22 (transversely orient) fat graph operad |
25 of the blob complex of the interval. |
23 \nn{need ref, or say more precisely what we mean}, |
|
24 and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
|
25 of the blob complex of the interval, thought of as a bimodule for itself. |
26 \nn{need to make sure we prove this above}. |
26 \nn{need to make sure we prove this above}. |
27 So the 1-dimensional Deligne conjecture can be restated as |
27 So the 1-dimensional Deligne conjecture can be restated as |
28 \[ |
28 \[ |
29 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
29 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
30 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
30 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
32 \] |
32 \] |
33 See Figure \ref{delfig1}. |
33 See Figure \ref{delfig1}. |
34 \begin{figure}[!ht] |
34 \begin{figure}[!ht] |
35 $$\mathfig{.9}{deligne/intervals}$$ |
35 $$\mathfig{.9}{deligne/intervals}$$ |
36 \caption{A fat graph}\label{delfig1}\end{figure} |
36 \caption{A fat graph}\label{delfig1}\end{figure} |
|
37 We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
|
38 for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
|
39 morphisms of such modules as defined in |
|
40 Subsection \ref{ss:module-morphisms}. |
37 |
41 |
38 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
42 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
39 of Figure \ref{delfig1} and ending at the topmost interval. |
43 of Figure \ref{delfig1} and ending at the topmost interval. |
40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
44 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
41 We remove the bottom interval of the bigon and replace it with the top interval. |
45 We remove the bottom interval of the bigon and replace it with the top interval. |
42 To map this topological operation to an algebraic one, we need, for each hole, an element of |
46 To convert this topological operation to an algebraic one, we need, for each hole, an element of |
43 $\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
47 $\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
44 So for each fixed fat graph we have a map |
48 So for each fixed fat graph we have a map |
45 \[ |
49 \[ |
46 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
50 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
47 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
51 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
55 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
52 |
56 |
53 It should now be clear how to generalize this to higher dimensions. |
57 It should now be clear how to generalize this to higher dimensions. |
54 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
58 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
55 involved were 1-dimensional. |
59 involved were 1-dimensional. |
56 Thus we can define a $n$-dimensional fat graph to be a sequence of general surgeries |
60 Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
57 on an $n$-manifold. |
61 on an $n$-manifold. |
|
62 |
|
63 \nn{*** resume revising here} |
|
64 |
58 More specifically, |
65 More specifically, |
59 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
66 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
60 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
67 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
61 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
68 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
62 (See Figure \ref{delfig2}.) |
69 (See Figure \ref{delfig2}.) |
63 \begin{figure}[!ht] |
70 \begin{figure}[!ht] |
64 $$\mathfig{.9}{deligne/manifolds}$$ |
71 $$\mathfig{.9}{deligne/manifolds}$$ |
65 \caption{A fat graph}\label{delfig2}\end{figure} |
72 \caption{A fat graph}\label{delfig2} |
|
73 \end{figure} |
|
74 |
|
75 |
|
76 |
|
77 |
|
78 |
|
79 |
66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
80 The components of the $n$-dimensional fat graph operad are indexed by tuples |
67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
81 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
68 \nn{not quite true: this is coarser than components} |
82 \nn{not quite true: this is coarser than components} |
69 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to |
83 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to |
70 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
84 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |