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1 \documentclass[11pt,leqno]{article} |
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2 |
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3 \usepackage{amsmath,amssymb,amsthm} |
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4 |
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5 \usepackage[all]{xy} |
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6 |
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7 |
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8 %%%%% excerpts from my include file of standard macros |
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9 |
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10 \def\bc{{\cal B}} |
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11 |
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12 \def\z{\mathbb{Z}} |
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13 \def\r{\mathbb{R}} |
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14 \def\c{\mathbb{C}} |
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15 \def\t{\mathbb{T}} |
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16 |
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17 \def\du{\sqcup} |
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18 \def\bd{\partial} |
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19 \def\sub{\subset} |
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20 \def\sup{\supset} |
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21 %\def\setmin{\smallsetminus} |
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22 \def\setmin{\setminus} |
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23 \def\ep{\epsilon} |
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24 \def\sgl{_\mathrm{gl}} |
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25 \def\deq{\stackrel{\mathrm{def}}{=}} |
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26 \def\pd#1#2{\frac{\partial #1}{\partial #2}} |
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27 |
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28 \def\nn#1{{{\it \small [#1]}}} |
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29 |
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30 |
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31 % equations |
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32 \newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}} |
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33 \newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}} |
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34 \newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}} |
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35 |
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36 % tricky way to iterate macros over a list |
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37 \def\semicolon{;} |
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38 \def\applytolist#1{ |
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39 \expandafter\def\csname multi#1\endcsname##1{ |
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40 \def\multiack{##1}\ifx\multiack\semicolon |
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41 \def\next{\relax} |
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42 \else |
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43 \csname #1\endcsname{##1} |
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44 \def\next{\csname multi#1\endcsname} |
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45 \fi |
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46 \next} |
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47 \csname multi#1\endcsname} |
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48 |
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49 % \def\cA{{\cal A}} for A..Z |
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50 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}} |
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51 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
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52 |
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53 % \DeclareMathOperator{\pr}{pr} etc. |
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54 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
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55 \applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign}; |
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56 |
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57 |
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58 |
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59 %%%%%% end excerpt |
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60 |
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61 |
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62 |
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63 |
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64 |
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65 \title{Blob Homology} |
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66 |
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67 \begin{document} |
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68 |
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69 |
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70 |
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71 \makeatletter |
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72 \@addtoreset{equation}{section} |
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73 \gdef\theequation{\thesection.\arabic{equation}} |
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74 \makeatother |
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75 \newtheorem{thm}[equation]{Theorem} |
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76 \newtheorem{prop}[equation]{Proposition} |
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77 \newtheorem{lemma}[equation]{Lemma} |
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78 \newtheorem{cor}[equation]{Corollary} |
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79 \newtheorem{defn}[equation]{Definition} |
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80 |
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81 |
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82 |
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83 \maketitle |
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84 |
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85 \section{Introduction} |
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86 |
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87 (motivation, summary/outline, etc.) |
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88 |
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89 (motivation: |
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90 (1) restore exactness in pictures-mod-relations; |
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91 (1') add relations-amongst-relations etc. to pictures-mod-relations; |
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92 (2) want answer independent of handle decomp (i.e. don't |
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93 just go from coend to derived coend (e.g. Hochschild homology)); |
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94 (3) ... |
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95 ) |
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96 |
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97 \section{Definitions} |
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98 |
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99 \subsection{Fields} |
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100 |
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101 Fix a top dimension $n$. |
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102 |
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103 A {\it system of fields} |
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104 \nn{maybe should look for better name; but this is the name I use elsewhere} |
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105 is a collection of functors $\cC$ from manifolds of dimension $n$ or less |
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106 to sets. |
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107 These functors must satisfy various properties (see KW TQFT notes for details). |
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108 For example: |
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109 there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; |
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110 there is a restriction map $\cC(X) \to \cC(\bd X)$; |
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111 gluing manifolds corresponds to fibered products of fields; |
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112 given a field $c \in \cC(Y)$ there is a ``product field" |
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113 $c\times I \in \cC(Y\times I)$; ... |
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114 \nn{should eventually include full details of definition of fields.} |
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115 |
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116 \nn{note: probably will suppress from notation the distinction |
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117 between fields and their (orientation-reversal) duals} |
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118 |
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119 \nn{remark that if top dimensional fields are not already linear |
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120 then we will soon linearize them(?)} |
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121 |
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122 The definition of a system of fields is intended to generalize |
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123 the relevant properties of the following two examples of fields. |
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124 |
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125 The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
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126 is a manifold of dimension $n$ or less) to be the set of |
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127 all maps from $X$ to $B$. |
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128 |
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129 The second example will take longer to explain. |
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130 Given an $n$-category $C$ with the right sort of duality |
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131 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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132 we can construct a system of fields as follows. |
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133 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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134 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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135 We'll spell this out for $n=1,2$ and then describe the general case. |
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136 |
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137 If $X$ has boundary, we require that the cell decompositions are in general |
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138 position with respect to the boundary --- the boundary intersects each cell |
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139 transversely, so cells meeting the boundary are mere half-cells. |
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140 |
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141 Put another way, the cell decompositions we consider are dual to standard cell |
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142 decompositions of $X$. |
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143 |
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144 We will always assume that our $n$-categories have linear $n$-morphisms. |
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145 |
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146 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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147 an object (0-morphism) of the 1-category $C$. |
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148 A field on a 1-manifold $S$ consists of |
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149 \begin{itemize} |
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150 \item A cell decomposition of $S$ (equivalently, a finite collection |
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151 of points in the interior of $S$); |
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152 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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153 by an object (0-morphism) of $C$; |
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154 \item a transverse orientation of each 0-cell, thought of as a choice of |
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155 ``domain" and ``range" for the two adjacent 1-cells; and |
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156 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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157 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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158 \end{itemize} |
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159 |
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160 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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161 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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162 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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163 of the algebra. |
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164 |
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165 For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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166 an object of the 2-category $C$. |
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167 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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168 A field on a 2-manifold $Y$ consists of |
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169 \begin{itemize} |
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170 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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171 that each component of the complement is homeomorphic to a disk); |
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172 \item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$) |
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173 by a 0-morphism of $C$; |
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174 \item a transverse orientation of each 1-cell, thought of as a choice of |
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175 ``domain" and ``range" for the two adjacent 2-cells; |
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176 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
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177 domain and range determined by the transverse orientation of the 1-cell |
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178 and the labelings of the 2-cells; |
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179 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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180 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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181 to $\pm 1 \in S^1$; and |
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182 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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183 determined by the labelings of the 1-cells and the parameterizations of the previous |
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184 bullet. |
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185 \end{itemize} |
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186 \nn{need to say this better; don't try to fit everything into the bulleted list} |
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187 |
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188 For general $n$, a field on a $k$-manifold $X^k$ consists of |
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189 \begin{itemize} |
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190 \item A cell decomposition of $X$; |
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191 \item an explicit general position homeomorphism from the link of each $j$-cell |
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192 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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193 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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194 domain and range determined by the labelings of the link of $j$-cell. |
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195 \end{itemize} |
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196 |
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197 \nn{next definition might need some work; I think linearity relations should |
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198 be treated differently (segregated) from other local relations, but I'm not sure |
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199 the next definition is the best way to do it} |
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200 |
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201 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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202 in the linearized space of fields. |
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203 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
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204 the vector space of finite |
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205 linear combinations of fields on $X$. |
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206 If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
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207 Thus the restriction (to boundary) maps are well defined because we never |
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208 take linear combinations of fields with differing boundary conditions. |
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209 |
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210 In some cases we don't linearize the default way; instead we take the |
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211 spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
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212 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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213 Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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214 obvious relations on 0-cell labels. |
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215 More specifically, let $L$ be a cell decomposition of $X$ |
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216 and let $p$ be a 0-cell of $L$. |
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217 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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218 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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219 Then the subspace $K$ is generated by things of the form |
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220 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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221 to infer the meaning of $\alpha_{\lambda c + d}$. |
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222 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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223 |
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224 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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225 will do something similar below; in general, whenever a label lives in a linear |
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226 space we do something like this; ? say something about tensor |
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227 product of all the linear label spaces? Yes:} |
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228 |
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229 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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230 Define an ``almost-field" to be a field without labels on the 0-cells. |
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231 (Recall that 0-cells are labeled by $n$-morphisms.) |
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232 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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233 space determined by the labeling of the link of the 0-cell. |
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234 (If the 0-cell were labeled, the label would live in this space.) |
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235 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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236 We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
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237 above tensor products. |
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238 |
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239 |
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240 |
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241 \subsection{Local relations} |
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242 |
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243 Let $B^n$ denote the standard $n$-ball. |
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244 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
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245 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
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246 |
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247 \nn{implies (extended?) isotopy; stable under gluing; open covers?; ...} |
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248 |
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249 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
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250 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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251 |
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252 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
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253 $\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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254 domain and range. |
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255 |
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256 \nn{maybe examples of local relations before general def?} |
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257 |
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258 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
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259 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
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260 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
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261 |
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262 Given a system of fields and local relations, we define the skein space |
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263 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
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264 the $n$-manifold $Y$ modulo local relations. |
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265 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
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266 is defined to be the dual of $A(Y; c)$. |
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267 (See KW TQFT notes or xxxx for details.) |
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268 |
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269 The blob complex is in some sense the derived version of $A(Y; c)$. |
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270 |
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271 |
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272 |
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273 \subsection{The blob complex} |
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274 |
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275 Let $X$ be an $n$-manifold. |
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276 Assume a fixed system of fields. |
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277 In this section we will usually suppress boundary conditions on $X$ from the notation |
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278 (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
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279 |
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280 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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281 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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282 $\overline{X \setmin Y}$. |
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283 |
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284 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
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285 |
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286 Define $\bc_0(X) = \cC_l(X)$. |
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287 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$. |
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288 We'll omit this sort of detail in the rest of this section.) |
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289 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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290 |
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291 $\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
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292 More specifically, define a 1-blob diagram to consist of |
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293 \begin{itemize} |
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294 \item An embedded closed ball (``blob") $B \sub X$. |
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295 %\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so. |
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296 %(See note in previous subsection.)} |
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297 %\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$. |
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298 \item A field $r \in \cC(X \setmin B; c)$ |
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299 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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300 \item A local relation field $u \in U(B; c)$ |
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301 (same $c$ as previous bullet). |
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302 \end{itemize} |
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303 %(Note that the the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
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304 %so we will omit $c$ from the notation.) |
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305 Define $\bc_1(X)$ to be the space of all finite linear combinations of |
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306 1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
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307 also the label ($u$ above) of the blob. |
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308 \nn{maybe spell this out in more detail} |
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309 (See xxxx above.) |
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310 \nn{maybe restate this in terms of direct sums of tensor products.} |
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311 |
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312 There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
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313 combination of fields on $X$ obtained by gluing $r$ to $u$. |
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314 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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315 just erasing the blob from the picture |
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316 (but keeping the blob label $u$). |
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317 |
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318 Note that the skein module $A(X)$ |
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319 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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320 |
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321 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
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322 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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323 2-blob diagrams (defined below), modulo the usual linear label relations. |
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324 \nn{and also modulo blob reordering relations?} |
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325 |
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326 \nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
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327 |
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328 There are two types of 2-blob diagram: disjoint and nested. |
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329 A disjoint 2-blob diagram consists of |
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330 \begin{itemize} |
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331 \item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$. |
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332 %\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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333 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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334 (where $c_i \in \cC(\bd B_i)$). |
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335 \item Local relation fields $u_i \in U(B_i; c_i)$. |
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336 \end{itemize} |
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337 Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$. |
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338 In other words, the boundary of a disjoint 2-blob diagram |
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339 is the sum (with alternating signs) |
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340 of the two ways of erasing one of the blobs. |
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341 It's easy to check that $\bd^2 = 0$. |
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342 |
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343 A nested 2-blob diagram consists of |
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344 \begin{itemize} |
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345 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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346 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
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347 (for some $c_0 \in \cC(\bd B_0)$). |
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348 Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
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349 (for some $c_1 \in \cC(B_1)$) and |
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350 $r' \in \cC(X \setmin B_1; c_1)$. |
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351 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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352 \end{itemize} |
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353 Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |
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354 Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$. |
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355 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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356 sum of the two ways of erasing one of the blobs. |
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357 If we erase the inner blob, the outer blob inherits the label $r_1u_0$. |
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358 |
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359 Now for the general case. |
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360 A $k$-blob diagram consists of |
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361 \begin{itemize} |
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362 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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363 For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or |
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364 $B_i \sub B_j$ or $B_j \sub B_i$. |
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365 (The case $B_i = B_j$ is allowed. |
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366 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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367 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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368 %\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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369 %(These are implied by the data in the next bullets, so we usually |
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370 %suppress them from the notation.) |
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371 %$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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372 %if the latter space is not empty. |
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373 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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374 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$. |
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375 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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376 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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377 If $B_i = B_j$ then $u_i = u_j$. |
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378 \end{itemize} |
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379 |
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380 We define $\bc_k(X)$ to be the vector space of all finite linear combinations |
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381 of $k$-blob diagrams, modulo the linear label relations and |
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382 blob reordering relations defined in the remainder of this paragraph. |
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383 Let $x$ be a blob diagram with one undetermined $n$-morphism label. |
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384 The unlabeled entity is either a blob or a 0-cell outside of the twig blobs. |
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385 Let $a$ and $b$ be two possible $n$-morphism labels for |
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386 the unlabeled blob or 0-cell. |
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387 Let $c = \lambda a + b$. |
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388 Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
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389 Then we impose the relation |
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390 \eq{ |
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391 x_c = \lambda x_a + x_b . |
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392 } |
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393 \nn{should do this in terms of direct sums of tensor products} |
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394 Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
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395 of their blob labelings. |
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396 Then we impose the relation |
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397 \eq{ |
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398 x = \sign(\pi) x' . |
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399 } |
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400 |
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401 (Alert readers will have noticed that for $k=2$ our definition |
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402 of $\bc_k(X)$ is slightly different from the previous definition |
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403 of $\bc_2(X)$. |
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404 The general definition takes precedence; |
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405 the earlier definition was simplified for purposes of exposition.) |
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406 |
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407 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
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408 Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
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409 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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410 If $B_j$ is not a twig blob, this involves only decrementing |
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411 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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412 If $B_j$ is a twig blob, we have to assign new local relation labels |
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413 if removing $B_j$ creates new twig blobs. |
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414 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
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415 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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416 Finally, define |
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417 \eq{ |
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418 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
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419 } |
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420 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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421 Thus we have a chain complex. |
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422 |
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423 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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424 |
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425 |
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426 \nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just |
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427 a vector space; relations to Chas-Sullivan string stuff} |
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428 |
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429 |
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430 |
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431 \section{Basic properties of the blob complex} |
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432 |
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433 \begin{prop} \label{disjunion} |
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434 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
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435 \end{prop} |
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436 \begin{proof} |
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437 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
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438 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
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439 blob diagram $(b_1, b_2)$ on $X \du Y$. |
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440 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
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441 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
442 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
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443 a pair of blob diagrams on $X$ and $Y$. |
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444 These two maps are compatible with our sign conventions \nn{say more about this?} and |
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445 with the linear label relations. |
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446 The two maps are inverses of each other. |
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447 \nn{should probably say something about sign conventions for the differential |
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448 in a tensor product of chain complexes; ask Scott} |
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449 \end{proof} |
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450 |
|
451 For the next proposition we will temporarily restore $n$-manifold boundary |
|
452 conditions to the notation. |
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453 |
|
454 Suppose that for all $c \in \cC(\bd B^n)$ |
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455 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
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456 of the quotient map |
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457 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
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458 \nn{always the case if we're working over $\c$}. |
|
459 Then |
|
460 \begin{prop} \label{bcontract} |
|
461 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
462 is a chain homotopy equivalence |
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463 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
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464 Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
465 \end{prop} |
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466 \begin{proof} |
|
467 By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
468 $h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
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469 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
470 an $(i{+}1)$-st blob equal to all of $B^n$. |
|
471 In other words, add a new outermost blob which encloses all of the others. |
|
472 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
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473 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
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474 \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
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475 \end{proof} |
|
476 |
|
477 (Note that for the above proof to work, we need the linear label relations |
|
478 for blob labels. |
|
479 Also we need to blob reordering relations (?).) |
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480 |
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481 (Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
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482 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
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483 |
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484 (For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.) |
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485 |
|
486 \medskip |
|
487 |
|
488 As we noted above, |
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489 \begin{prop} |
|
490 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
491 \qed |
|
492 \end{prop} |
|
493 |
|
494 |
|
495 \begin{prop} |
|
496 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
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497 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
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498 (chain map) isomorphisms. |
|
499 \qed |
|
500 \end{prop} |
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501 |
|
502 |
|
503 In particular, |
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504 \begin{prop} \label{diff0prop} |
|
505 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
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506 \qed |
|
507 \end{prop} |
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508 |
|
509 The above will be greatly strengthened in Section \ref{diffsect}. |
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510 |
|
511 \medskip |
|
512 |
|
513 For the next proposition we will temporarily restore $n$-manifold boundary |
|
514 conditions to the notation. |
|
515 |
|
516 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
517 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
518 with boundary $Z\sgl$. |
|
519 Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
|
520 we have the blob complex $\bc_*(X; a, b, c)$. |
|
521 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
522 $X$ to get blob diagrams on $X\sgl$: |
|
523 |
|
524 \begin{prop} |
|
525 There is a natural chain map |
|
526 \eq{ |
|
527 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
|
528 } |
|
529 The sum is over all fields $a$ on $Y$ compatible at their |
|
530 ($n{-}2$-dimensional) boundaries with $c$. |
|
531 `Natural' means natural with respect to the actions of diffeomorphisms. |
|
532 \qed |
|
533 \end{prop} |
|
534 |
|
535 The above map is very far from being an isomorphism, even on homology. |
|
536 This will be fixed in Section \ref{gluesect} below. |
|
537 |
|
538 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
|
539 and $X\sgl = X_1 \cup_Y X_2$. |
|
540 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
541 For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
542 \eq{ |
|
543 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
|
544 } |
|
545 Note that we have resumed our habit of omitting boundary labels from the notation. |
|
546 |
|
547 |
|
548 \bigskip |
|
549 |
|
550 \nn{what else?} |
|
551 |
|
552 |
|
553 |
|
554 |
|
555 \section{$n=1$ and Hochschild homology} |
|
556 |
|
557 In this section we analyze the blob complex in dimension $n=1$ |
|
558 and find that for $S^1$ the homology of the blob complex is the |
|
559 Hochschild homology of the category (algebroid) that we started with. |
|
560 |
|
561 Notation: $HB_i(X) = H_i(\bc_*(X))$. |
|
562 |
|
563 Let us first note that there is no loss of generality in assuming that our system of |
|
564 fields comes from a category. |
|
565 (Or maybe (???) there {\it is} a loss of generality. |
|
566 Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
|
567 thought of as the morphisms of a 1-category $C$. |
|
568 More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
|
569 are $A(I; a, b)$, and composition is given by gluing. |
|
570 If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
|
571 and neither does $A(I; a, b) = HB_0(I; a, b)$. |
|
572 But what about $HB_i(I; a, b)$ for $i > 0$? |
|
573 Might these higher blob homology groups be different? |
|
574 Seems unlikely, but I don't feel like trying to prove it at the moment. |
|
575 In any case, we'll concentrate on the case of fields based on 1-category |
|
576 pictures for the rest of this section.) |
|
577 |
|
578 (Another question: $\bc_*(I)$ is an $A_\infty$-category. |
|
579 How general of an $A_\infty$-category is it? |
|
580 Given an arbitrary $A_\infty$-category can one find fields and local relations so |
|
581 that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
|
582 Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
|
583 |
|
584 Continuing... |
|
585 |
|
586 Let $C$ be a *-1-category. |
|
587 Then specializing the definitions from above to the case $n=1$ we have: |
|
588 \begin{itemize} |
|
589 \item $\cC(pt) = \ob(C)$ . |
|
590 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
|
591 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
|
592 points in the interior |
|
593 of $R$, each labeled by a morphism of $C$. |
|
594 The intervals between the points are labeled by objects of $C$, consistent with |
|
595 the boundary condition $c$ and the domains and ranges of the point labels. |
|
596 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
|
597 composing the morphism labels of the points. |
|
598 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
|
599 point (at some standard location) labeled by $x$. |
|
600 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
|
601 form $y - \chi(e(y))$. |
|
602 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
|
603 \end{itemize} |
|
604 |
|
605 We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
|
606 Hochschild homology of $C$. |
|
607 \nn{Or better that the complexes are homotopic |
|
608 or quasi-isomorphic.} |
|
609 In order to prove this we will need to extend the blob complex to allow points to also |
|
610 be labeled by elements of $C$-$C$-bimodules. |
|
611 %Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
|
612 %(over $C$) of $C$-$C$-bimodules. |
|
613 %Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
|
614 %Now we can define the blob complex for $S^1$. |
|
615 %This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
|
616 %If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
|
617 %to the cyclic 1-tuple $(M)$. |
|
618 %In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
|
619 %by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
|
620 %It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
|
621 %blob complex for $S^1$. |
|
622 %\nn{Is it really so clear? Should say more.} |
|
623 |
|
624 %\nn{alternative to the above paragraph:} |
|
625 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
|
626 We define a blob-like complex $F_*(S^1, (p_i), (M_i))$. |
|
627 The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
|
628 other points. |
|
629 The blob twig labels lie in kernels of evaluation maps. |
|
630 (The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
|
631 Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
|
632 In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$ |
|
633 and elements of $C$ at variable other points. |
|
634 |
|
635 We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild |
|
636 homology of $M$. |
|
637 \nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend. |
|
638 Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
|
639 complex of $M$.} |
|
640 This follows from the following lemmas: |
|
641 \begin{itemize} |
|
642 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
|
643 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ |
|
644 gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$. |
|
645 (See below for proof.) |
|
646 \item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
|
647 homotopic to the 0-step complex $C$. |
|
648 (See below for proof.) |
|
649 \item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is homotopic to $\bc_*(S^1)$. |
|
650 (See below for proof.) |
|
651 \end{itemize} |
|
652 |
|
653 First we show that $F_*(C\otimes C)$ is |
|
654 homotopic to the 0-step complex $C$. |
|
655 |
|
656 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
|
657 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
|
658 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
|
659 |
|
660 Fix a small $\ep > 0$. |
|
661 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
|
662 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex where $B_\ep$ is either disjoint from |
|
663 or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points. |
|
664 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
|
665 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
|
666 (See Figure xxxx.) |
|
667 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
|
668 |
|
669 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
|
670 Let $x \in F^\ep_*$ be a blob diagram. |
|
671 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
|
672 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
|
673 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
|
674 Let $y_i$ be the restriction of $z_i$ to $*$. |
|
675 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
|
676 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
|
677 Define $j_\ep(x) = \sum x_i$. |
|
678 |
|
679 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
|
680 |
|
681 The key property of $j_\ep$ is |
|
682 \eq{ |
|
683 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
|
684 } |
|
685 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction of each field |
|
686 mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$. |
|
687 Note that $\sigma_\ep(x) \in F'$. |
|
688 |
|
689 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
|
690 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
|
691 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
|
692 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
|
693 Instead, we'll be less ambitious and just show that |
|
694 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
|
695 |
|
696 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ |
|
697 $x \in F_*^\ep$. |
|
698 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
|
699 finitely many blob diagrams.) |
|
700 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
|
701 is surjective on homology. |
|
702 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
|
703 and |
|
704 \eq{ |
|
705 \bd x = \bd (\sigma_\ep(y) + j_\ep(x)) . |
|
706 } |
|
707 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
|
708 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
|
709 |
|
710 \medskip |
|
711 |
|
712 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
|
713 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
|
714 |
|
715 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
|
716 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
|
717 Then $G''_*$ and $G'_*$ are both contractible. |
|
718 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
|
719 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
|
720 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
|
721 in ``basic properties" section above} away from $*$. |
|
722 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
|
723 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
|
724 Actually, we need the further (easy) result that the inclusion |
|
725 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
|
726 |
|
727 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
|
728 for all $x \in F'_*$ we have |
|
729 \eq{ |
|
730 x - \bd h(x) - h(\bd x) \in F''_* . |
|
731 } |
|
732 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
|
733 Let $x \in F'_1$, with single blob $B \sub S^1$. |
|
734 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
|
735 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
|
736 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
|
737 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
|
738 Define $h_1(x) = y$. |
|
739 The general case is similar, except that we have to take lower order homotopies into account. |
|
740 Let $x \in F'_k$. |
|
741 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
|
742 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
|
743 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
|
744 So $x' \in G'_l$ for some $l \le k$. |
|
745 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
|
746 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
|
747 Define $h_k(x) = y \bullet p$. |
|
748 This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence. |
|
749 \nn{need to say above more clearly and settle on notation/terminology} |
|
750 |
|
751 Finally, we show that $F''_*$ is contractible. |
|
752 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
|
753 Let $x$ be a cycle in $F''_*$. |
|
754 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
|
755 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
|
756 Adding $B$ as a blob to $x$ gives a contraction. |
|
757 \nn{need to say something else in degree zero} |
|
758 |
|
759 This completes the proof that $F_*(C\otimes C)$ is |
|
760 homotopic to the 0-step complex $C$. |
|
761 |
|
762 \medskip |
|
763 |
|
764 Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$ |
|
765 \nn{...} |
|
766 |
|
767 \bigskip |
|
768 |
|
769 \nn{still need to prove exactness claim} |
|
770 |
|
771 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
|
772 Do we need a map from hoch to blob? |
|
773 Does the above exactness and contractibility guarantee such a map without writing it |
|
774 down explicitly? |
|
775 Probably it's worth writing down an explicit map even if we don't need to.} |
|
776 |
|
777 |
|
778 |
|
779 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
|
780 |
|
781 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
|
782 the space of diffeomorphisms |
|
783 of the $n$-manifold $X$ (fixed on $\bd X$). |
|
784 For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
|
785 than simplices --- they can be based on any linear polyhedron. |
|
786 \nn{be more restrictive here? does more need to be said?} |
|
787 |
|
788 \begin{prop} \label{CDprop} |
|
789 For each $n$-manifold $X$ there is a chain map |
|
790 \eq{ |
|
791 e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
|
792 } |
|
793 On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
|
794 (Proposition (\ref{diff0prop})). |
|
795 For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
|
796 \eq{ \xymatrix{ |
|
797 CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
|
798 CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
799 \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
|
800 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
|
801 } } |
|
802 Any other map satisfying the above two properties is homotopic to $e_X$. |
|
803 \end{prop} |
|
804 |
|
805 The proof will occupy the remainder of this section. |
|
806 |
|
807 \medskip |
|
808 |
|
809 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
|
810 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
|
811 $x \notin S$ and $p, q \in P$. |
|
812 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
|
813 |
|
814 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
815 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
|
816 {\it adapted to $\cU$} if there is a factorization |
|
817 \eq{ |
|
818 P = P_1 \times \cdots \times P_m |
|
819 } |
|
820 (for some $m \le k$) |
|
821 and families of diffeomorphisms |
|
822 \eq{ |
|
823 f_i : P_i \times X \to X |
|
824 } |
|
825 such that |
|
826 \begin{itemize} |
|
827 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
|
828 \item the $V_i$'s are mutually disjoint; |
|
829 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
|
830 where $k_i = \dim(P_i)$; and |
|
831 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
|
832 for all $p = (p_1, \ldots, p_m)$. |
|
833 \end{itemize} |
|
834 A chain $x \in C_k(\Diff(M))$ is (by definition) adapted to $\cU$ if is is the sum |
|
835 of singular cells, each of which is adapted to $\cU$. |
|
836 |
|
837 \begin{lemma} \label{extension_lemma} |
|
838 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
839 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
840 \end{lemma} |
|
841 |
|
842 The proof will be given in Section \ref{fam_diff_sect}. |
|
843 |
|
844 \medskip |
|
845 |
|
846 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
|
847 (e.g.~the support of a blob diagram). |
|
848 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
|
849 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
|
850 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
|
851 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
|
852 each of which is compatible. |
|
853 (Note that we could strengthen the definition of compatibility to incorporate |
|
854 a factorization condition, similar to the definition of ``adapted to" above. |
|
855 The weaker definition given here will suffice for our needs below.) |
|
856 |
|
857 \begin{cor} \label{extension_lemma_2} |
|
858 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
|
859 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
|
860 \end{cor} |
|
861 \begin{proof} |
|
862 This will follow from Lemma \ref{extension_lemma} for |
|
863 appropriate choice of cover $\cU = \{U_\alpha\}$. |
|
864 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
|
865 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
|
866 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
|
867 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
|
868 |
|
869 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
|
870 with the (compatible) support of $f_i$ in place of $X$. |
|
871 This insures that the resulting homotopy $h_i$ is compatible. |
|
872 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
|
873 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
|
874 \end{proof} |
|
875 |
|
876 |
|
877 |
|
878 |
|
879 \section{Families of Diffeomorphisms} \label{fam_diff_sect} |
|
880 |
|
881 |
|
882 Lo, the proof of Lemma (\ref{extension_lemma}): |
|
883 |
|
884 \nn{should this be an appendix instead?} |
|
885 |
|
886 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
887 later draft} |
|
888 |
|
889 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
890 about boundary later} |
|
891 |
|
892 Recall that we are given |
|
893 an open cover $\cU = \{U_\alpha\}$ and an |
|
894 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
895 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
896 |
|
897 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
898 |
|
899 As a first approximation to the argument we will eventually make, let's replace $x$ |
|
900 with a single singular cell |
|
901 \eq{ |
|
902 f: P \times X \to X . |
|
903 } |
|
904 Also, we'll ignore for now issues around $\bd P$. |
|
905 |
|
906 Our homotopy will have the form |
|
907 \eqar{ |
|
908 F: I \times P \times X &\to& X \\ |
|
909 (t, p, x) &\mapsto& f(u(t, p, x), x) |
|
910 } |
|
911 for some function |
|
912 \eq{ |
|
913 u : I \times P \times X \to P . |
|
914 } |
|
915 First we describe $u$, then we argue that it does what we want it to do. |
|
916 |
|
917 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
918 The various $K_\alpha$ should be in general position with respect to each other. |
|
919 We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
920 to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
921 merely a homotopy through maps $X\to X$. |
|
922 |
|
923 Let $L$ be the union of all the $K_\alpha$'s. |
|
924 $L$ is itself a cell decomposition of $P$. |
|
925 \nn{next two sentences not needed?} |
|
926 To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
927 where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
928 Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
929 |
|
930 Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
931 Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
932 more importantly, a $k{-}i$-dimensional normal coordinate. |
|
933 |
|
934 For each $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
|
935 Let $D$ be a $k$-handle of $J$, and let $d$ also denote the corresponding |
|
936 $k$-cell of $L$. |
|
937 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
938 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
939 |
|
940 For $p \in D$ we define |
|
941 \eq{ |
|
942 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
|
943 } |
|
944 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
945 makes sense.) |
|
946 |
|
947 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
948 For handles of $J$ of index less than $k$, we will define $u$ to |
|
949 interpolate between the values on $k$-handles defined above. |
|
950 |
|
951 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
|
952 of $E$. |
|
953 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
954 with a $k$-handle. |
|
955 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
956 corresponding to $E$. |
|
957 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
958 adjacent to the $k{-}1$-cell corresponding to $E$. |
|
959 For $p \in E$, define |
|
960 \eq{ |
|
961 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
|
962 + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
963 } |
|
964 |
|
965 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
966 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
967 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
968 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
969 a map from the the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
|
970 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
|
971 the $k{-}j$-cell corresponding to $E$. |
|
972 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
973 Now define, for $p \in E$, |
|
974 \eq{ |
|
975 u(t, p, x) = (1-t)p + t \left( |
|
976 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
977 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
978 \right) . |
|
979 } |
|
980 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
981 mentioned above. |
|
982 |
|
983 This completes the definition of $u: I \times P \times X \to P$. |
|
984 |
|
985 \medskip |
|
986 |
|
987 Next we verify that $u$ has the desired properties. |
|
988 |
|
989 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
|
990 Therefore $F$ is a homotopy from $f$ to something. |
|
991 |
|
992 Next we show that the the $K_\alpha$'s are sufficiently fine cell decompositions, |
|
993 then $F$ is a homotopy through diffeomorphisms. |
|
994 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
995 We have |
|
996 \eq{ |
|
997 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
|
998 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
999 } |
|
1000 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
1001 \nn{bounded away from zero, or something like that}. |
|
1002 (Recall that $X$ and $P$ are compact.) |
|
1003 Also, $\pd{f}{p}$ is bounded. |
|
1004 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
1005 It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
1006 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
|
1007 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
1008 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
1009 |
|
1010 \medskip |
|
1011 |
|
1012 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1013 is a singular cell adapted to $\cU$. |
|
1014 This will complete the proof of the lemma. |
|
1015 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
1016 |
|
1017 Let $j$ be the codimension of $D$. |
|
1018 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
|
1019 between handle and corresponding cell.) |
|
1020 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
1021 where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
1022 cells of codimension greater than 0 which intersect to form $D$. |
|
1023 We will show that |
|
1024 if the relevant $U_\alpha$'s are disjoint, then |
|
1025 $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1026 is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
1027 If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
1028 cells whose dimensions correspond to a partition of the $j_i$'s. |
|
1029 We will consider some simple special cases first, then do the general case. |
|
1030 |
|
1031 First consider the case $j=0$ (and $m=0$). |
|
1032 A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
1033 is independent of $p \in P$. |
|
1034 So the corresponding map $D \to \Diff(X)$ is constant. |
|
1035 |
|
1036 Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
1037 Now Equation yyyy applies. |
|
1038 We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
1039 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
1040 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
1041 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
1042 |
|
1043 Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
1044 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
|
1045 1-dimensional. |
|
1046 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
1047 and a 2-cell with support $U_\beta$. |
|
1048 |
|
1049 Next case: $j=2$, $m=2$, $j_1 = j_2 = 2$. |
|
1050 In this case the codimension 2 cell $D$ is the intersection of two |
|
1051 codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
1052 We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
1053 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
1054 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1055 two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
1056 If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1057 a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
1058 \nn{need to check that this is true} |
|
1059 |
|
1060 \nn{finally, general case...} |
|
1061 |
|
1062 \nn{this completes proof} |
|
1063 |
|
1064 |
|
1065 |
|
1066 |
|
1067 \section{$A_\infty$ action on the boundary} |
|
1068 |
|
1069 |
|
1070 \section{Gluing} \label{gluesect} |
|
1071 |
|
1072 \section{Extension to ...} |
|
1073 |
|
1074 (Need to let the input $n$-category $C$ be a graded thing |
|
1075 (e.g.~DGA or $A_\infty$ $n$-category).) |
|
1076 |
|
1077 |
|
1078 \section{What else?...} |
|
1079 |
|
1080 \begin{itemize} |
|
1081 \item Derive Hochschild standard results from blob point of view? |
|
1082 \item $n=2$ examples |
|
1083 \item Kh |
|
1084 \item dimension $n+1$ |
|
1085 \item should be clear about PL vs Diff; probably PL is better |
|
1086 (or maybe not) |
|
1087 \item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
|
1088 \item something about higher derived coend things (derived 2-coend, e.g.) |
|
1089 \end{itemize} |
|
1090 |
|
1091 |
|
1092 |
|
1093 \end{document} |
|
1094 |
|
1095 |
|
1096 |
|
1097 %Recall that for $n$-category picture fields there is an evaluation map |
|
1098 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
1099 %If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
1100 %map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |
|
1101 |
|
1102 |
|
1103 |