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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Introduction} |
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4 |
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5 [Outline for intro] |
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6 \begin{itemize} |
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7 \item Starting point: TQFTs via fields and local relations. |
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8 This gives a satisfactory treatment for semisimple TQFTs |
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9 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
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10 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
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11 \item For non-semiemple TQFTs, this approach is less satisfactory. |
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12 Our main motivating example (though we will not develop it in this paper) |
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13 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
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14 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
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15 with a link $L \subset \bd W$. |
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16 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
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17 \item How would we go about computing $A_{Kh}(W^4, L)$? |
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18 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
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19 \nn{... $L_1, L_2, L_3$}. |
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20 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
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21 to compute $A_{Kh}(S^1\times B^3, L)$. |
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22 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
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23 corresponds to taking a coend (self tensor product) over the cylinder category |
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24 associated to $B^3$ (with appropriate boundary conditions). |
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25 The coend is not an exact functor, so the exactness of the triangle breaks. |
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26 \item The obvious solution to this problem is to replace the coend with its derived counterpart. |
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27 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
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28 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
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29 If we build our manifold up via a handle decomposition, the computation |
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30 would be a sequence of derived coends. |
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31 A different handle decomposition of the same manifold would yield a different |
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32 sequence of derived coends. |
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33 To show that our definition in terms of derived coends is well-defined, we |
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34 would need to show that the above two sequences of derived coends yield the same answer. |
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35 This is probably not easy to do. |
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36 \item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
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37 which is manifestly invariant. |
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38 (That is, a definition that does not |
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39 involve choosing a decomposition of $W$. |
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40 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
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41 is that it has just this sort of manifest invariance.) |
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42 \item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
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43 \[ |
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44 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
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45 \] |
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46 with an appropriately free resolution (the ``blob complex") |
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47 \[ |
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48 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
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49 \] |
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50 Here $\bc_0$ is linear combinations of fields on $W$, |
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51 $\bc_1$ is linear combinations of local relations on $W$, |
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52 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
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53 and so on. |
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54 \item None of the above ideas depend on the details of the Khovanov homology example, |
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55 so we develop the general theory in the paper and postpone specific applications |
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56 to later papers. |
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57 \item The blob complex enjoys the following nice properties \nn{...} |
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58 \end{itemize} |
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59 |
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60 \bigskip |
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61 \hrule |
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62 \bigskip |
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63 |
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64 We then show that blob homology enjoys the following |
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65 \ref{property:gluing} properties. |
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66 |
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67 \begin{property}[Functoriality] |
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68 \label{property:functoriality}% |
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69 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
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70 \begin{equation*} |
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71 X \mapsto \bc_*^{\cF,\cU}(X) |
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72 \end{equation*} |
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73 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
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74 \end{property} |
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75 |
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76 \begin{property}[Disjoint union] |
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77 \label{property:disjoint-union} |
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78 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
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79 \begin{equation*} |
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80 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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81 \end{equation*} |
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82 \end{property} |
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83 |
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84 \begin{property}[A map for gluing] |
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85 \label{property:gluing-map}% |
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86 If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, |
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87 there is a chain map |
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88 \begin{equation*} |
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89 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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90 \end{equation*} |
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91 \end{property} |
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92 |
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93 \begin{property}[Contractibility] |
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94 \label{property:contractibility}% |
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95 \todo{Err, requires a splitting?} |
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96 The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
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97 \begin{equation} |
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98 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
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99 \end{equation} |
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100 \todo{Say that this is just the original $n$-category?} |
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101 \end{property} |
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102 |
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103 \begin{property}[Skein modules] |
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104 \label{property:skein-modules}% |
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105 The $0$-th blob homology of $X$ is the usual |
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106 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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107 by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.) |
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108 \begin{equation*} |
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109 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
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110 \end{equation*} |
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111 \end{property} |
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112 |
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113 \begin{property}[Hochschild homology when $X=S^1$] |
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114 \label{property:hochschild}% |
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115 The blob complex for a $1$-category $\cC$ on the circle is |
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116 quasi-isomorphic to the Hochschild complex. |
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117 \begin{equation*} |
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118 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
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119 \end{equation*} |
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120 \end{property} |
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121 |
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122 \begin{property}[Evaluation map] |
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123 \label{property:evaluation}% |
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124 There is an `evaluation' chain map |
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125 \begin{equation*} |
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126 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
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127 \end{equation*} |
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128 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
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129 |
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130 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
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131 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
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132 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
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133 \begin{equation*} |
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134 \xymatrix{ |
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135 \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
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136 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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137 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
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138 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
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139 } |
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140 \end{equation*} |
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141 \nn{should probably say something about associativity here (or not?)} |
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142 \end{property} |
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143 |
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144 |
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145 \begin{property}[Gluing formula] |
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146 \label{property:gluing}% |
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147 \mbox{}% <-- gets the indenting right |
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148 \begin{itemize} |
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149 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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150 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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151 |
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152 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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153 $A_\infty$ module for $\bc_*(Y \times I)$. |
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154 |
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155 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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156 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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157 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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158 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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159 \begin{equation*} |
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160 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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161 \end{equation*} |
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162 \end{itemize} |
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163 \end{property} |
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164 |
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165 \nn{add product formula? $n$-dimensional fat graph operad stuff?} |
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166 |
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167 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
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168 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
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169 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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170 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
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171 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |