9 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
9 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
10 \end{property:deligne} |
10 \end{property:deligne} |
11 |
11 |
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 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. |
13 |
13 |
14 \def\mapinf{\Maps_\infty} |
14 %\def\mapinf{\Maps_\infty} |
15 |
15 |
16 The usual Deligne conjecture \nn{need refs} gives a map |
16 The usual Deligne conjecture \nn{need refs} gives a map |
17 \[ |
17 \[ |
18 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
18 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
19 \to Hoch^*(C, C) . |
19 \to Hoch^*(C, C) . |
23 The little disks operad is homotopy equivalent to the fat graph operad |
23 The little disks operad is homotopy equivalent to the fat graph operad |
24 \nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
24 \nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
25 of the blob complex of the interval. |
25 of the blob complex of the interval. |
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 \begin{eqnarray*} |
28 \[ |
29 C_*(FG_k)\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
29 C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
30 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\ |
30 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
31 & \hspace{-5em} \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
31 \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
32 \end{eqnarray*} |
32 \] |
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 |
37 |
38 We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
38 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. |
39 of Figure \ref{delfig1} and ending at the topmost interval. |
40 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
40 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. |
41 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, element of |
42 To map this topological operation to an algebraic one, we need, for each hole, an element of |
43 $\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
43 $\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
44 So for each fixed fat graph we have a map |
44 So for each fixed fat graph we have a map |
45 \[ |
45 \[ |
46 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
46 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
47 \otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) \to \mapinf(\bc^C_*(I), \bc^C_*(I)) . |
47 \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
48 \] |
48 \] |
49 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
49 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
50 between the maps associated to the endpoints of the 1-chain. |
50 between the maps associated to the endpoints of the 1-chain. |
51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
51 Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
52 |
52 |
63 \begin{figure}[!ht] |
63 \begin{figure}[!ht] |
64 $$\mathfig{.9}{deligne/manifolds}$$ |
64 $$\mathfig{.9}{deligne/manifolds}$$ |
65 \caption{A fat graph}\label{delfig2}\end{figure} |
65 \caption{A fat graph}\label{delfig2}\end{figure} |
66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
68 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to |
68 \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 |
69 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
70 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
|
71 \nn{what about rotating in the horizontal directions?} |
70 |
72 |
71 |
73 |
72 If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define |
74 If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define |
73 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
75 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
74 $A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both |
76 $A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both |
80 Putting this together we get |
82 Putting this together we get |
81 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
83 \begin{prop}(Precise statement of Property \ref{property:deligne}) |
82 \label{prop:deligne} |
84 \label{prop:deligne} |
83 There is a collection of maps |
85 There is a collection of maps |
84 \begin{eqnarray*} |
86 \begin{eqnarray*} |
85 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
87 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
86 \mapinf(\bc_*(M_{k}), \bc_*(N_{k})) & \\ |
88 \hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\ |
87 & \hspace{-11em}\to \mapinf(\bc_*(M_0), \bc_*(N_0)) |
89 & \hspace{-11em}\to \hom(\bc_*(M_0), \bc_*(N_0)) |
88 \end{eqnarray*} |
90 \end{eqnarray*} |
89 which satisfy an operad type compatibility condition. \nn{spell this out} |
91 which satisfy an operad type compatibility condition. \nn{spell this out} |
90 \end{prop} |
92 \end{prop} |
91 |
93 |
92 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |
94 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |