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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Gluing - needs to be rewritten/replaced} |
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4 \label{sec:gluing}% |
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5 |
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6 \nn{*** this section is now obsolete; should be removed soon} |
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7 |
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8 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
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9 \begin{itemize} |
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10 %\mbox{}% <-- gets the indenting right |
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11 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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12 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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13 |
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14 \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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15 $A_\infty$ module for $\bc_*(Y \times I)$. |
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16 |
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17 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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18 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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19 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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20 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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21 \begin{equation*} |
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22 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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23 \end{equation*} |
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24 \end{itemize} |
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25 |
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26 Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
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27 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
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28 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
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29 the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
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30 as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
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31 Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
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32 self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs |
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33 of the gluing formula in the topological context. |
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34 |
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35 \subsection{`Topological' $A_\infty$ $n$-categories} |
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36 \label{sec:topological-A-infty}% |
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37 |
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38 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
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39 The main result of this section is |
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40 |
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41 \begin{thm} |
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42 Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
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43 $A_\infty$-$1$-categories. |
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44 \end{thm} |
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45 |
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46 Before proving this theorem, we embark upon a long string of definitions. |
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47 For expository purposes, we begin with the $n=1$ special cases, |
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48 and define |
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49 first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
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50 to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
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51 \nn{Something about duals?} |
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52 \todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
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53 \nn{probably we should say something about the relation |
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54 to [framed] $E_\infty$ algebras |
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55 } |
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56 |
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57 \todo{} |
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58 Various citations we might want to make: |
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59 \begin{itemize} |
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60 \item \cite{MR2061854} McClure and Smith's review article |
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61 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
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62 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
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63 \item \cite{MR1256989} definition of framed little-discs operad |
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64 \end{itemize} |
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65 |
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66 \begin{defn} |
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67 \label{defn:topological-algebra}% |
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68 A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
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69 \begin{enumerate} |
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70 \item For each $1$-manifold $J$ diffeomorphic to the standard interval |
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71 $I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
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72 % either roll functoriality into the evaluation map |
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73 \item For each pair of intervals $J,J'$ an `evaluation' chain map |
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74 $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
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75 \item For each decomposition of intervals $J = J'\cup J''$, |
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76 a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
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77 % or do it as two separate pieces of data |
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78 %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
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79 %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
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80 %\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
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81 \end{enumerate} |
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82 This data is required to satisfy the following conditions. |
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83 \begin{itemize} |
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84 \item The evaluation chain map is associative, in that the diagram |
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85 \begin{equation*} |
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86 \xymatrix{ |
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87 & \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\ |
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88 \CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\ |
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89 & A(J'') & |
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90 } |
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91 \end{equation*} |
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92 commutes up to homotopy. |
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93 Here the map $$\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$$ is a composition: take products of singular chains first, then compose diffeomorphisms. |
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94 %% or the version for separate pieces of data: |
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95 %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
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96 %\item The evaluation chain map is associative, in that the diagram |
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97 %\begin{equation*} |
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98 %\xymatrix{ |
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99 %\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
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100 %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
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101 %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
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102 %A(J) |
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103 %} |
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104 %\end{equation*} |
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105 %commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
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106 \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
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107 \begin{equation*} |
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108 \xymatrix{ |
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109 A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && |
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110 A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
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111 A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
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112 A(J \cup J' \cup J'') |
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113 } |
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114 \end{equation*} |
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115 commutes. |
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116 \item The gluing and evaluation maps are compatible. |
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117 \nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
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118 \end{itemize} |
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119 \end{defn} |
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120 |
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121 \begin{rem} |
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122 We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
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123 constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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124 Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
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125 \end{rem} |
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126 |
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127 %% if we do things separately, we should say this: |
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128 %\begin{rem} |
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129 %Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
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130 %intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
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131 %Further, one can combine the second and third pieces of data, asking instead for a map |
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132 %\begin{equation*} |
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133 %\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
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134 %\end{equation*} |
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135 %(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
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136 %diffeomorphisms in $\CD{J'}$.) |
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137 %\end{rem} |
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138 |
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139 To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
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140 interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
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141 \begin{equation*} |
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142 \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
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143 \end{equation*} |
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144 The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
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145 \todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
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146 |
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147 At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
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148 \begin{defn} |
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149 Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
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150 \begin{enumerate} |
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151 \item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
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152 \item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
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153 \begin{align*} |
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154 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
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155 \end{align*} |
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156 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
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157 \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
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158 \end{enumerate} |
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159 The associativity conditions are trivially satisfied. |
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160 \end{defn} |
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161 |
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162 The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
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163 Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties |
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164 \ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
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165 |
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166 The definition of a module follows closely the definition of an algebra or category. |
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167 \begin{defn} |
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168 \label{defn:topological-module}% |
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169 A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
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170 consists of the following data. |
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171 \begin{enumerate} |
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172 \item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces. |
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173 \item For each pair of such marked intervals, |
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174 an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
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175 \item For each decomposition $K = J\cup K'$ of the marked interval |
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176 $K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
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177 $\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
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178 \end{enumerate} |
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179 The above data is required to satisfy |
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180 conditions analogous to those in Definition \ref{defn:topological-algebra}. |
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181 \end{defn} |
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182 |
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183 For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as |
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184 a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
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185 For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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186 (Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
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187 \ref{property:evaluation} and \ref{property:gluing-map} respectively. |
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188 |
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189 The definition of a bimodule is like the definition of a module, |
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190 except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
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191 on the upper boundary and the other with a marked point on the lower boundary. |
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192 There are evaluation maps corresponding to gluing unmarked intervals |
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193 to the unmarked ends of $K$ and $L$. |
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194 |
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195 Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
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196 codimension-0 submanifold of $\bdy X$. |
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197 Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
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198 structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
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199 |
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200 Next we define the coend |
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201 (or gluing or tensor product or self tensor product, depending on the context) |
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202 $\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
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203 \begin{defn} |
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204 We define a category $\cG(M)$. Objects consist of the following data. |
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205 \begin{itemize} |
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206 \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
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207 \item For each pair of intervals $N,N'$ an evaluation chain map |
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208 $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
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209 \item For each decomposition of intervals $N = K\cup L$, |
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210 a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
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211 \end{itemize} |
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212 This data must satisfy the following conditions. |
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213 \begin{itemize} |
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214 \item The evaluation maps are associative. |
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215 \nn{up to homotopy?} |
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216 \item Gluing is strictly associative. |
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217 That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
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218 $K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
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219 agree. |
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220 \item the gluing and evaluation maps are compatible. |
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221 \end{itemize} |
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222 |
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223 A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked, |
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224 satisfying the following conditions. |
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225 \begin{itemize} |
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226 \item For each pair of intervals $N,N'$, the diagram |
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227 \begin{equation*} |
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228 \xymatrix{ |
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229 \CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\ |
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230 C(N) \ar[r]_{f_N} & C'(N) |
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231 } |
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232 \end{equation*} |
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233 commutes. |
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234 \item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition |
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235 $$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
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236 \end{itemize} |
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237 \end{defn} |
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238 |
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239 We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$, |
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240 there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$ |
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241 factor through the gluing maps for $\gl(M)$. |
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242 |
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243 We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself |
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244 is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
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245 |
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246 For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
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247 $$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
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248 Showing that it is an initial object is the content of the gluing theorem proved below. |
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249 |
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250 |
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251 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
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252 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
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253 easy, I think, so maybe it should be done earlier??} |
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254 |
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255 \bigskip |
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256 |
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257 Outline: |
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258 \begin{itemize} |
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259 \item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
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260 use graphical/tree point of view, rather than following Keller exactly |
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261 \item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
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262 \item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
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263 \item motivating example: $C_*(\Maps(X, M))$ |
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264 \item maybe incorporate dual point of view (for $n=1$), where points get |
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265 object labels and intervals get 1-morphism labels |
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266 \end{itemize} |
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267 |
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268 |
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269 \subsection{$A_\infty$ action on the boundary} |
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270 \label{sec:boundary-action}% |
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271 Let $Y$ be an $n{-}1$-manifold. |
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272 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
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273 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
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274 of an $A_\infty$ category. |
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275 |
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276 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
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277 $I\cup I \cong I$. Given this choice, gluing gives a map |
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278 \eq{ |
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279 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
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280 \cong \bc_*(Y\times I; a, c) |
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281 } |
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282 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
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283 higher associators of the $A_\infty$ structure, more or less canonically. |
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284 |
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285 \nn{is this obvious? does more need to be said?} |
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286 |
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287 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
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288 |
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289 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
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290 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
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291 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
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292 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
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293 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
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294 of $Y$ in $X$. |
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295 |
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296 In the next section we use the above $A_\infty$ actions to state and prove |
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297 a gluing theorem for the blob complexes of $n$-manifolds. |
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298 |
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299 |
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300 \subsection{The gluing formula} |
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301 \label{sec:gluing-formula}% |
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302 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
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303 of $Y \du -Y$ contained in its boundary. |
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304 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
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305 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
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306 of $X$. |
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307 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
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308 where $c\sgl \in \cC(\bd X\sgl)$, |
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309 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
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310 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
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311 |
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312 \begin{thm} |
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313 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
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314 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
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315 \end{thm} |
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316 |
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317 The proof will occupy the remainder of this section. |
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318 |
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319 \nn{...} |
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320 |
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321 \bigskip |
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322 |
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323 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
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324 |