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1 % use options |
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2 % '[beamer]' for a digital projector |
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3 % '[trans]' for an overhead projector |
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4 % '[handout]' for 4-up printed notes |
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5 \documentclass[beamer, compress]{beamer} |
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6 |
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7 % change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. |
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8 \newcommand{\pathtotrunk}{../../} |
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9 \input{\pathtotrunk talks/talk_preamble.tex} |
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10 |
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11 %\setbeameroption{previous slide on second screen=right} |
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12 |
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13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} |
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14 \institute{UC Berkeley / Miller Institute for Basic Research} |
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15 \title{Blob homology, part $\mathbb{I}$} |
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16 \date{Homotopy Theory and Higher Algebraic Structures, UC Riverside, November 10 2009 \\ \begin{description}\item[slides, part $\mathbb{I}$:]\url{http://tqft.net/UCR-blobs1} \item[slides, part $\mathbb{II}$:]\url{http://tqft.net/UCR-blobs2} \item[draft:]\url{http://tqft.net/blobs}\end{description}} |
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17 |
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18 \begin{document} |
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19 |
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20 \frame{\titlepage} |
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21 |
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22 \beamertemplatetransparentcovered |
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23 |
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24 \mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}} |
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25 |
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26 \beamersetuncovermixins |
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27 {\opaqueness<1->{60}} |
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28 {} |
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29 |
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30 |
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31 |
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32 \section{Overview} |
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33 |
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34 \begin{frame}<beamer> |
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35 \frametitle{Blob homology} |
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36 \begin{quote} |
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37 ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. |
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38 \end{quote} |
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39 \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} |
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40 \tableofcontents |
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41 \end{frame} |
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42 |
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43 \begin{frame}{What is \emph{blob homology}?} |
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44 \begin{block}{} |
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45 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
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46 \end{block} |
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47 \tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm] |
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48 \begin{tikzpicture}[] |
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49 \setbeamercovered{% |
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50 transparent=5, |
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51 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
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52 again covered={\opaqueness<1->{50}} |
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53 } |
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54 |
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55 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
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56 \uncover<1>{ |
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57 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
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58 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
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59 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
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60 } |
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61 |
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62 \uncover<2>{ |
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63 \node[blue] (hoch) at (0,3) {$HH_*(\cC)$}; |
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64 \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)}; |
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65 \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch); |
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66 } |
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67 |
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68 \uncover<3>{ |
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69 \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$}; |
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70 \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)}; |
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71 \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm); |
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72 } |
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73 |
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74 \end{tikzpicture} |
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75 \end{frame} |
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76 |
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77 \begin{frame}{$n$-categories} |
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78 \begin{block}{Defining $n$-categories is fraught with difficulties} |
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79 I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. |
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80 \end{block} |
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81 \begin{block}{} |
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82 Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. |
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83 \end{block} |
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84 |
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85 \begin{block}{} |
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86 \begin{itemize} |
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87 \item |
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88 Defining $n$-categories: a choice of `shape' for morphisms. |
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89 \item |
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90 We allow all shapes! A vector space for every ball. |
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91 \item |
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92 `Strong duality' is integral in our definition. |
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93 \end{itemize} |
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94 \end{block} |
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95 \end{frame} |
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96 |
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97 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
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98 |
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99 \section{Definition} |
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100 \begin{frame}{Fields and pasting diagrams} |
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101 \begin{block}{Pasting diagrams} |
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102 Fix an $n$-category with strong duality $\cC$. A \emph{field} on $\cM$ is a pasting diagram drawn on $\cM$, with cells labelled by morphisms from $\cC$. |
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103 \end{block} |
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104 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
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106 \end{example} |
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107 \begin{block}{} |
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108 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
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109 \vspace{-3mm} |
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110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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111 \end{block} |
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112 \end{frame} |
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113 |
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114 \begin{frame}{Background: TQFT invariants} |
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115 \begin{defn} |
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116 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |
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117 \end{defn} |
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118 (`decapitated': no numerical invariants of $n+1$-manifolds.) |
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119 |
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120 \begin{block}{} |
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121 If the manifold has boundary, we get a category. Objects are boundary data, $\Hom{\cA(\cM)}{x}{y} = \cA(\cM; x,y)$. |
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122 \end{block} |
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123 |
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124 \begin{block}{} |
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125 We want to extend `all the way down'. The $k$-category associated to the $n-k$-manifold $\cY$ is $\cA(\cY \times B^k)$. |
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126 \end{block} |
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127 |
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128 \begin{defn} |
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129 Given an $n$-category $\cC$, the associated TQFT is |
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130 \vspace{-3mm} |
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131 $$\cA(\cM) = \cF(M) / \ker{ev},$$ |
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132 |
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133 \vspace{-3mm} |
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134 fields modulo fields which evaluate to zero inside some ball. |
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135 \end{defn} |
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136 \end{frame} |
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137 |
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138 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
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139 \begin{block}{Motivation} |
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140 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. |
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141 \end{block} |
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142 |
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143 \begin{block}{} |
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144 \center |
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145 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
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146 \end{block} |
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147 |
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148 \begin{block}{} |
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149 \vspace{-1mm} |
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150 $$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
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151 \end{block} |
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152 \vspace{-3.5mm} |
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153 $$\mathfig{.5}{definition/single-blob}$$ |
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154 \vspace{-3mm} |
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155 \begin{block}{} |
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156 \vspace{-6mm} |
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157 \begin{align*} |
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158 d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) |
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159 \end{align*} |
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160 \end{block} |
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161 \end{frame} |
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162 |
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163 \begin{frame}{Definition, $k=2$} |
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164 \begin{block}{} |
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165 \vspace{-1mm} |
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166 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
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167 \end{block} |
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168 \begin{block}{} |
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169 \vspace{-5mm} |
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170 \begin{align*} |
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171 \bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
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172 \end{align*} |
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173 \vspace{-4mm} |
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174 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
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175 \end{block} |
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176 \begin{block}{} |
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177 \vspace{-5mm} |
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178 \begin{align*} |
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179 \bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} |
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180 \end{align*} |
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181 \vspace{-4mm} |
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182 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
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183 \end{block} |
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184 \end{frame} |
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185 |
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186 \begin{frame}{Definition, general case} |
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187 \begin{block}{} |
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188 $$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ |
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189 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
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190 \end{block} |
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191 \begin{block}{} |
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192 \vspace{-2mm} |
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193 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
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194 \end{block} |
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195 \end{frame} |
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196 |
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197 \section{Properties} |
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198 \begin{frame}{Hochschild homology} |
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199 \begin{block}{TQFT on $S^1$ is `coinvariants'} |
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200 \vspace{-3mm} |
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201 $$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$ |
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202 \end{block} |
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203 \begin{block}{} |
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204 The Hochschild complex is `coinvariants of the bar resolution' |
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205 \vspace{-2mm} |
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206 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
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207 \end{block} |
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208 \begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$] |
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209 $$m \tensor a \mapsto |
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210 \roundframe{\mathfig{0.35}{hochschild/1-chains}} |
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211 $$ |
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212 \vspace{-5mm} |
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213 \begin{align*} |
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214 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 &= \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
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215 \end{align*} |
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216 \end{thm} |
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217 \end{frame} |
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218 |
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219 \begin{frame}{An action of $\CH{\cM}$} |
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220 \begin{thm} |
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221 There's a chain map |
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222 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
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223 which is associative up to homotopy, and compatible with gluing. |
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224 \end{thm} |
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225 \begin{block}{} |
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226 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
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227 \end{block} |
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228 \end{frame} |
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229 |
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230 \begin{frame}{Higher Deligne conjecture} |
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231 \begin{block}{Deligne conjecture} |
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232 Chains on the little discs operad acts on Hochschild cohomology. |
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233 \end{block} |
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234 |
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235 \begin{block}{} |
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236 Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'. |
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237 \end{block} |
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238 |
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239 \begin{block}{Theorem* (Higher Deligne conjecture)} |
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240 \scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.} |
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241 \vspace{-3mm} |
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242 $$\mathfig{.85}{deligne/manifolds}$$ |
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243 \end{block} |
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244 \end{frame} |
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245 |
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246 \begin{frame}{Gluing} |
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247 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
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248 \begin{itemize} |
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249 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
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250 \item[$m_k$:] reparametrising $[0,1]$ |
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251 \end{itemize} |
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252 \end{block} |
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253 \begin{block}{} |
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254 If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. |
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255 \end{block} |
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256 \begin{thm}[Gluing formula] |
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257 When $Y \sqcup Y^{\text{op}} \subset \bdy X$, |
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258 \vspace{-5mm} |
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259 \[ |
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260 \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. |
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261 \] |
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262 \end{thm} |
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263 In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. |
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264 \end{frame} |
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265 \end{document} |
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266 % ---------------------------------------------------------------- |
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267 |