53 It should now be clear how to generalize this to higher dimensions. |
53 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 |
54 In the sequence-of-surgeries description above, we never used the fact that the manifolds |
55 involved were 1-dimensional. |
55 involved were 1-dimensional. |
56 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries |
56 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries |
57 on an $n$-manifold. |
57 on an $n$-manifold. |
58 More specifically, \nn{...} |
58 More specifically, |
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59 the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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60 $R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
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61 $f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
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62 (See Figure \ref{delfig2}.) |
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63 \begin{figure}[!ht] |
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64 $$\mathfig{.9}{tempkw/delfig2}$$ |
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65 \caption{A fat graph}\label{delfig2}\end{figure} |
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66 The components of the $n$-dimensional fat graph operad are indexed by tuples |
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67 $(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
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68 Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to |
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69 the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
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70 |
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71 |
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72 If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define |
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73 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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74 $A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both |
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75 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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76 The ``holes" in the above |
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77 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
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78 \nn{need to make up my mind which notation I'm using for the module maps} |
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79 |
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80 Putting this together we get a collection of maps |
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81 \begin{eqnarray*} |
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82 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes |
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83 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
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84 & \hspace{-11em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
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85 \end{eqnarray*} |
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86 which satisfy an operad type compatibility condition. |
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87 |
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88 Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |
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89 And indeed, the proof is very similar \nn{...} |
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90 |
59 |
91 |
60 |
92 |
61 \medskip |
93 \medskip |
62 \hrule\medskip |
94 \hrule\medskip |
63 |
95 |
64 |
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65 Figure \ref{delfig2} |
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66 \begin{figure}[!ht] |
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67 $$\mathfig{.9}{tempkw/delfig2}$$ |
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68 \caption{A fat graph}\label{delfig2}\end{figure} |
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69 |
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70 |
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71 \begin{eqnarray*} |
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72 C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes |
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73 \mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\ |
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74 & \hspace{-5em}\to \mapinf(\bc_*(M_k), \bc_*(N_k)) |
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75 \end{eqnarray*} |
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76 |
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77 \medskip |
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78 \hrule\medskip |
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79 |
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80 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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81 of $n$-manifolds |
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82 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
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83 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
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84 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
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85 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
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86 |
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87 If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define |
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88 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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89 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
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90 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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91 The ``holes" in the above |
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92 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
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