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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{The blob complex} |
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4 \label{sec:blob-definition} |
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5 |
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6 Let $X$ be an $n$-manifold. |
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7 Assume a fixed system of fields and local relations. |
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8 In this section we will usually suppress boundary conditions on $X$ from the notation |
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9 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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10 |
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11 We want to replace the quotient |
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12 \[ |
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13 A(X) \deq \lf(X) / U(X) |
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14 \] |
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15 of the previous section with a resolution |
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16 \[ |
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17 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
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18 \] |
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19 |
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20 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
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21 |
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22 We of course define $\bc_0(X) = \lf(X)$. |
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23 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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24 We'll omit this sort of detail in the rest of this section.) |
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25 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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26 |
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27 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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28 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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29 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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30 \begin{itemize} |
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31 \item An embedded closed ball (``blob") $B \sub X$. |
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32 \item A field $r \in \cC(X \setmin B; c)$ |
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33 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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34 \item A local relation field $u \in U(B; c)$ |
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35 (same $c$ as previous bullet). |
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36 \end{itemize} |
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37 (See Figure \ref{blob1diagram}.) |
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38 \begin{figure}[!ht]\begin{equation*} |
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39 \mathfig{.9}{definition/single-blob} |
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40 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
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41 In order to get the linear structure correct, we (officially) define |
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42 \[ |
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43 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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44 \] |
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45 The first direct sum is indexed by all blobs $B\subset X$, and the second |
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46 by all boundary conditions $c \in \cC(\bd B)$. |
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47 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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48 |
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49 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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50 \[ |
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51 (B, u, r) \mapsto u\bullet r, |
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52 \] |
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53 where $u\bullet r$ denotes the linear |
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54 combination of fields on $X$ obtained by gluing $u$ to $r$. |
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55 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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56 just erasing the blob from the picture |
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57 (but keeping the blob label $u$). |
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58 |
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59 Note that the skein space $A(X)$ |
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60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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61 |
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62 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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63 local relations encoded in $\bc_1(X)$. |
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64 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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65 2-blob diagrams, of which there are two types, disjoint and nested. |
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66 |
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67 A disjoint 2-blob diagram consists of |
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68 \begin{itemize} |
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69 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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70 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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71 (where $c_i \in \cC(\bd B_i)$). |
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72 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
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73 \end{itemize} |
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74 (See Figure \ref{blob2ddiagram}.) |
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75 \begin{figure}[!ht]\begin{equation*} |
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76 \mathfig{.9}{definition/disjoint-blobs} |
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77 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
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78 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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79 reversing the order of the blobs changes the sign. |
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80 Define $\bd(B_0, B_1, u_0, u_1, r) = |
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81 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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82 In other words, the boundary of a disjoint 2-blob diagram |
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83 is the sum (with alternating signs) |
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84 of the two ways of erasing one of the blobs. |
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85 It's easy to check that $\bd^2 = 0$. |
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86 |
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87 A nested 2-blob diagram consists of |
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88 \begin{itemize} |
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89 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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90 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
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91 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
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92 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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93 \end{itemize} |
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94 (See Figure \ref{blob2ndiagram}.) |
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95 \begin{figure}[!ht]\begin{equation*} |
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96 \mathfig{.9}{definition/nested-blobs} |
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97 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
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98 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
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99 (for some $c_1 \in \cC(B_1)$) and |
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100 $r' \in \cC(X \setmin B_1; c_1)$. |
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101 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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102 Note that the requirement that |
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103 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
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104 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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105 sum of the two ways of erasing one of the blobs. |
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106 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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107 It is again easy to check that $\bd^2 = 0$. |
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108 |
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109 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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110 (officially) |
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111 \begin{eqnarray*} |
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112 \bc_2(X) & \deq & |
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113 \left( |
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114 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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115 U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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116 \right) \\ |
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117 && \bigoplus \left( |
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118 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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119 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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120 \right) . |
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121 \end{eqnarray*} |
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122 The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, |
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123 but we didn't feel like introducing a notation for that. |
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124 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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125 (rather than a new, linearly independent 2-blob diagram). |
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126 |
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127 Now for the general case. |
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128 A $k$-blob diagram consists of |
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129 \begin{itemize} |
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130 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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131 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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132 $B_i \sub B_j$ or $B_j \sub B_i$. |
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133 (The case $B_i = B_j$ is allowed. |
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134 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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135 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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136 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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137 (These are implied by the data in the next bullets, so we usually |
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138 suppress them from the notation.) |
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139 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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140 if the latter space is not empty. |
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141 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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142 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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143 is determined by the $c_i$'s. |
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144 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
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145 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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146 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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147 If $B_i = B_j$ then $u_i = u_j$. |
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148 \end{itemize} |
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149 (See Figure \ref{blobkdiagram}.) |
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150 \begin{figure}[!ht]\begin{equation*} |
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151 \mathfig{.9}{definition/k-blobs} |
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152 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
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153 |
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154 If two blob diagrams $D_1$ and $D_2$ |
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155 differ only by a reordering of the blobs, then we identify |
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156 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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157 |
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158 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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159 As before, the official definition is in terms of direct sums |
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160 of tensor products: |
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161 \[ |
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162 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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163 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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164 \] |
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165 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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166 $\overline{c}$ runs over all boundary conditions, again as described above. |
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167 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
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168 |
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169 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
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170 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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171 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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172 If $B_j$ is not a twig blob, this involves only decrementing |
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173 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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174 If $B_j$ is a twig blob, we have to assign new local relation labels |
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175 if removing $B_j$ creates new twig blobs. |
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176 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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177 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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178 Finally, define |
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179 \eq{ |
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180 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
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181 } |
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182 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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183 Thus we have a chain complex. |
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184 |
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185 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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186 |
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187 \nn{?? remark about dendroidal sets?; probably not} |
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188 |
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189 |