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1 In this section we analyze the blob complex in dimension $n=1$ |
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2 and find that for $S^1$ the homology of the blob complex is the |
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3 Hochschild homology of the category (algebroid) that we started with. |
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4 \nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
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5 should perhaps put more emphasis on the complexes and less on the homology.} |
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6 |
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7 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
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8 |
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9 Let us first note that there is no loss of generality in assuming that our system of |
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10 fields comes from a category. |
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11 (Or maybe (???) there {\it is} a loss of generality. |
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12 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
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13 thought of as the morphisms of a 1-category $C$. |
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14 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
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15 are $A(I; a, b)$, and composition is given by gluing. |
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16 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
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17 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
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18 But what about $HB_i(I; a, b)$ for $i > 0$? |
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19 Might these higher blob homology groups be different? |
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20 Seems unlikely, but I don't feel like trying to prove it at the moment. |
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21 In any case, we'll concentrate on the case of fields based on 1-category |
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22 pictures for the rest of this section.) |
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23 |
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24 (Another question: $\bc_*(I)$ is an $A_\infty$-category. |
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25 How general of an $A_\infty$-category is it? |
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26 Given an arbitrary $A_\infty$-category can one find fields and local relations so |
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27 that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
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28 Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
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29 |
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30 Continuing... |
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31 |
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32 Let $C$ be a *-1-category. |
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33 Then specializing the definitions from above to the case $n=1$ we have: |
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34 \begin{itemize} |
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35 \item $\cC(pt) = \ob(C)$ . |
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36 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
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37 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
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38 points in the interior |
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39 of $R$, each labeled by a morphism of $C$. |
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40 The intervals between the points are labeled by objects of $C$, consistent with |
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41 the boundary condition $c$ and the domains and ranges of the point labels. |
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42 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
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43 composing the morphism labels of the points. |
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44 Note that we also need the * of *-1-category here in order to make all the morphisms point |
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45 the same way. |
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46 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
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47 point (at some standard location) labeled by $x$. |
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48 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
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49 form $y - \chi(e(y))$. |
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50 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
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51 \end{itemize} |
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52 |
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53 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
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54 Hochschild homology of $C$. |
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55 \nn{Or better that the complexes are homotopic |
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56 or quasi-isomorphic.} |
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57 In order to prove this we will need to extend the blob complex to allow points to also |
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58 be labeled by elements of $C$-$C$-bimodules. |
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59 %Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
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60 %(over $C$) of $C$-$C$-bimodules. |
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61 %Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
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62 %Now we can define the blob complex for $S^1$. |
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63 %This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
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64 %If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
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65 %to the cyclic 1-tuple $(M)$. |
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66 %In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
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67 %by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
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68 %It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
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69 %blob complex for $S^1$. |
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70 %\nn{Is it really so clear? Should say more.} |
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71 |
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72 %\nn{alternative to the above paragraph:} |
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73 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
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74 We define a blob-like complex $F_*(S^1, (p_i), (M_i))$. |
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75 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
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76 other points. |
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77 The blob twig labels lie in kernels of evaluation maps. |
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78 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
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79 Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
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80 In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$ |
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81 and elements of $C$ at variable other points. |
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82 |
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83 \todo{Some orphaned questions:} |
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84 \nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend. |
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85 Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
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86 complex of $M$.} |
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87 |
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88 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
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89 Do we need a map from hoch to blob? |
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90 Does the above exactness and contractibility guarantee such a map without writing it |
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91 down explicitly? |
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92 Probably it's worth writing down an explicit map even if we don't need to.} |
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93 |
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94 |
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95 We claim that |
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96 \begin{thm} |
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97 The blob complex $\bc_*(S^1; C)$ on the circle is quasi-isomorphic to the |
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98 usual Hochschild complex for $C$. |
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99 \end{thm} |
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100 |
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101 This follows from two results. First, we see that |
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102 \begin{lem} |
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103 \label{lem:module-blob}% |
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104 The complex $F_*(C)$ (here $C$ is being thought of as a |
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105 $C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex |
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106 $\bc_*(S^1; C)$. (Proof later.) |
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107 \end{lem} |
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108 |
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109 Next, we show that for any $C$-$C$-bimodule $M$, |
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110 \begin{prop} |
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111 The complex $F_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual |
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112 Hochschild complex of $M$. |
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113 \end{prop} |
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114 \begin{proof} |
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115 First, since we're working over $\Complex$, note that saying two complexes are quasi-isomorphic simply means they have isomorphic homologies. |
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116 \todo{We really don't want to work over $\Complex$, though; it would be nice to talk about torsion!} |
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117 |
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118 Recall that the usual Hochschild complex of $M$ is uniquely determined, up to quasi-isomorphism, by the following properties: |
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119 \begin{enumerate} |
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120 \item \label{item:hochschild-additive}% |
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121 $HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$. |
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122 \item \label{item:hochschild-exact}% |
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123 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
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124 exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. |
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125 \item \label{item:hochschild-free}% |
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126 $HC_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
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127 quasi-isomorphic to the 0-step complex $C$. |
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128 \item \label{item:hochschild-coinvariants}% |
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129 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
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130 \end{enumerate} |
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131 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
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132 We'll first explain why these properties are characteristic. Take some |
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133 $C$-$C$ bimodule $M$. If $M$ is free, that is, a direct sum of copies of |
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134 $C \tensor C$, then properties \ref{item:hochschild-additive} and |
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135 \ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some |
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136 free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
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137 have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
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138 short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) |
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139 \to 0$. Such a sequence gives a long exact sequence on homology |
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140 \begin{equation*} |
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141 %\begin{split} |
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142 \cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ |
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143 %\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). |
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144 %\end{split} |
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145 \end{equation*} |
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146 For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties |
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147 \ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
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148 $HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. |
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149 |
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150 This tells us how to |
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151 compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
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152 (it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
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153 and higher homology groups are determined by lower ones in $HC_*(K)$, and |
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154 hence recursively as coinvariants of some other bimodule. |
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155 |
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156 The proposition then follows from the following lemmas, establishing that $F_*$ has precisely these required properties. |
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157 \begin{lem} |
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158 \label{lem:hochschild-additive}% |
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159 Directly from the definition, $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
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160 \end{lem} |
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161 \begin{lem} |
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162 \label{lem:hochschild-exact}% |
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163 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
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164 exact sequence $0 \to F_*(M_1) \into F_*(M_2) \onto F_*(M_3) \to 0$. |
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165 \end{lem} |
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166 \begin{lem} |
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167 \label{lem:hochschild-free}% |
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168 $F_*(C\otimes C)$ is quasi-isomorphic to the 0-step complex $C$. |
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169 \end{lem} |
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170 \begin{lem} |
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171 \label{lem:hochschild-coinvariants}% |
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172 $H_0(F_*(M))$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
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173 \end{lem} |
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174 |
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175 The remainder of this section is devoted to proving Lemmas |
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176 \ref{lem:module-blob}, |
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177 \ref{lem:hochschild-exact}, \ref{lem:hochschild-free} and |
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178 \ref{lem:hochschild-coinvariants}. |
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179 \end{proof} |
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180 |
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181 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
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182 We show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
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183 $F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
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184 is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
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185 In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$. |
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186 |
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187 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
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188 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
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189 * is a labeled point in $y$. |
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190 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
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191 Let $x \in \bc_*(S^1)$. |
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192 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
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193 $x$ with $y$. |
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194 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
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195 |
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196 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
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197 in a neighborhood $B_\ep$ of *, except perhaps *. |
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198 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
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199 \nn{rest of argument goes similarly to above} |
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200 \end{proof} |
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201 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
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202 \todo{} |
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203 \end{proof} |
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204 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
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205 We show that $F_*(C\otimes C)$ is |
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206 quasi-isomorphic to the 0-step complex $C$. |
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207 |
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208 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
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209 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
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210 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
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211 |
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212 Fix a small $\ep > 0$. |
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213 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
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214 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
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215 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
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216 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
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217 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
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218 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
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219 (See Figure xxxx.) |
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220 Note that $y - s_\ep(y) \in U(B_\ep)$. |
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221 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
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222 |
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223 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
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224 Let $x \in F^\ep_*$ be a blob diagram. |
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225 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
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226 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
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227 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
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228 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
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229 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
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230 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
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231 Define $j_\ep(x) = \sum x_i$. |
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232 \nn{need to check signs coming from blob complex differential} |
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233 |
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234 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
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235 |
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236 The key property of $j_\ep$ is |
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237 \eq{ |
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238 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
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239 } |
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240 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
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241 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
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242 Note that $\sigma_\ep(x) \in F'_*$. |
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243 |
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244 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
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245 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
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246 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
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247 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
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248 Instead, we'll be less ambitious and just show that |
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249 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
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250 |
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251 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
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252 $x \in F_*^\ep$. |
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253 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
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254 finitely many blob diagrams.) |
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255 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
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256 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
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257 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
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258 and |
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259 \eq{ |
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260 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
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261 } |
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262 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
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263 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
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264 |
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265 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
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266 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
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267 |
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268 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
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269 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
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270 Then $G''_*$ and $G'_*$ are both contractible |
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271 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
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272 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
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273 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
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274 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
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275 in ``basic properties" section above} away from $*$. |
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276 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
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277 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
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278 Actually, we need the further (easy) result that the inclusion |
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279 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
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280 |
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281 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
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282 for all $x \in F'_*$ we have |
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283 \eq{ |
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284 x - \bd h(x) - h(\bd x) \in F''_* . |
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285 } |
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286 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
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287 Let $x \in F'_1$, with single blob $B \sub S^1$. |
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288 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
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289 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
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290 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
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291 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
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292 Define $h_1(x) = y$. |
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293 The general case is similar, except that we have to take lower order homotopies into account. |
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294 Let $x \in F'_k$. |
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295 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
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296 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
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297 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
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298 So $x' \in G'_l$ for some $l \le k$. |
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299 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
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300 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
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301 Define $h_k(x) = y \bullet p$. |
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302 This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence. |
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303 \nn{need to say above more clearly and settle on notation/terminology} |
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304 |
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305 Finally, we show that $F''_*$ is contractible. |
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306 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
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307 Let $x$ be a cycle in $F''_*$. |
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308 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
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309 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
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310 Adding $B$ as a blob to $x$ gives a contraction. |
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311 \nn{need to say something else in degree zero} |
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312 \end{proof} |
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313 \begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] |
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314 \todo{} |
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315 \end{proof} |
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316 |
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317 We can also describe explicitly a map from the standard Hochschild |
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318 complex to the blob complex on the circle. \nn{What properties does this |
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319 map have?} |
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320 |
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321 \begin{figure}% |
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322 $$\mathfig{0.6}{barycentric/barycentric}$$ |
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323 \caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
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324 the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
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325 \label{fig:Hochschild-example}% |
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326 \end{figure} |
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327 |
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328 As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. |
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329 The edges marked $x, y$ and $z$ carry the $1$-chains |
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330 \begin{align*} |
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331 x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
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332 y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
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333 z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
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334 \end{align*} |
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335 and the $2$-chain labelled $A$ is |
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336 \begin{equation*} |
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337 A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
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338 \end{equation*} |
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339 Note that we then have |
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340 \begin{equation*} |
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341 \bdy A = x+y+z. |
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342 \end{equation*} |
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343 |
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344 In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, |
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345 $$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
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346 with ... (hmmm, problems making this precise; you need to decide where to put the labels, but then it's hard to make an honest chain map!) |