75 } |
75 } |
76 |
76 |
77 \end{tikzpicture} |
77 \end{tikzpicture} |
78 \end{frame} |
78 \end{frame} |
79 |
79 |
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80 \begin{frame}{$n$-categories} |
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81 \begin{block}{There are many definitions of $n$-categories!} |
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82 For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal $2$-categories. |
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83 \end{block} |
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84 \begin{block}{We have another definition!} |
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85 \emph{Many axioms}; geometric examples are easy, algebraic ones hard. |
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86 \begin{itemize} |
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87 %\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
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88 \item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
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89 %\item From these, inductively |
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90 %\begin{itemize} |
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91 %\item define a set $\cC(S^k)$ for each $k$-sphere, $0 \leq k < n$, |
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92 %\item require a map $\cC(B^k) \to \cC(S^{k-1})$. |
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93 %\end{itemize} |
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94 \item An associative gluing map: with $B = \bigcup_i B_i$, balls glued together to form a ball, |
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95 $$\bigotimes \cC(B_i) \to \cC(B)$$ |
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96 (the $\tensor$ is fibered over `boundary restriction' maps). |
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97 \item ... |
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98 \end{itemize} |
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99 \end{block} |
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100 \end{frame} |
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101 |
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102 \begin{frame}{Cellulations of manifolds} |
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103 \begin{block}{} |
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104 Consider $\cell(M)$, the category of cellulations of a manifold $M$, with morphisms `antirefinements'. |
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105 \end{block} |
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106 \vspace{-4mm} |
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107 $$\mathfig{.35}{ncat/zz2}$$ |
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108 \vspace{-4mm} |
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109 \begin{block}{} |
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110 An $n$-category $\cC$ gives a functor from $\cell(M)$ to vector spaces. |
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111 \begin{description} |
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112 \item[objects] send a cellulation to the product of $\cC$ on each top-cell, restricting to the subset where boundaries agree |
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113 \item[morphisms] send an antirefinement to the appropriate gluing map. |
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114 \end{description} |
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115 \end{block} |
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116 \end{frame} |
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117 |
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118 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
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119 |
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120 \section{Definition} |
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121 \begin{frame}{Fields} |
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122 \begin{block}{} |
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123 A field on $\cM^n$ is a choice of cellulation and a choice of $n$-morphism for each top-cell. |
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124 %$$\cF(\cM) = \bigoplus_{\cX \in \cell(M)} \bigotimes_{B \in \cX} \cC(B)$$ |
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125 \end{block} |
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126 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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127 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
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128 \end{example} |
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129 \begin{block}{} |
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130 Given a field on a ball, we can evaluate it to a morphism using the gluing map. We call the kernel the \emph{null fields}. |
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131 \vspace{-3mm} |
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132 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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133 \end{block} |
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134 \end{frame} |
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135 |
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136 \begin{frame}{Background: TQFT invariants} |
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137 \begin{defn} |
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138 We associate to an $n$-manifold $\cM$ the skein module |
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139 \vspace{-1mm} |
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140 $$\cA(\cM) = \cF(\cM) / \ker{ev},\vspace{-1mm}$$ |
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141 fields modulo fields which evaluate to zero inside some ball. |
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142 \end{defn} |
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143 Equivalently, $\cA(\cM)$ is the colimit of $\cC$ along $\cell(M)$. |
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144 |
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145 \vspace{4mm} |
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146 %\begin{itemize} |
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147 %\item We can also associate a $k$-category to an $n-k$-manifold. |
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148 %\item We don't assign a number to an $n+1$-manifold (a `decapitated' extended TQFT). |
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149 %\end{itemize} |
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150 $\cA(Y \times [0,1])$ is a $1$-category, and when $Y \subset \bdy X$, $\cA(X)$ is a module over $\cA(Y \times [0,1])$. |
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151 \begin{thm}[Gluing formula] |
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152 When $Y \sqcup Y^{\text{op}} \subset \bdy X$, |
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153 \vspace{-1mm} |
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154 \[ |
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155 \cA(X \bigcup_Y \selfarrow) \iso \cA(X) \bigotimes_{\cA(Y \times [0,1])} \selfarrow. |
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156 \] |
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157 \end{thm} |
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158 \end{frame} |
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159 |
80 \begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} |
160 \begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} |
81 \begin{thm} |
161 \begin{thm} |
82 Khovanov homology gives a $4$-category: |
162 Khovanov homology gives a $4$-category: |
83 \begin{description} |
163 \begin{description} |
84 \item[3-morphisms] tangles, with the usual $3$ operations, |
164 \item[3-morphisms] tangles, with the usual $3$ operations, |
91 \end{block} |
171 \end{block} |
92 \end{frame} |
172 \end{frame} |
93 |
173 |
94 \begin{frame}{Computations are hard} |
174 \begin{frame}{Computations are hard} |
95 \begin{block}{} |
175 \begin{block}{} |
96 The corresponding $4$-manifold invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. |
176 This invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. |
97 \vspace{-0.3cm} |
177 \vspace{-0.3cm} |
98 \begin{align*} |
178 \begin{align*} |
99 \begin{tikzpicture} |
179 \begin{tikzpicture} |
100 \node(a) at (0,0) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
180 \node(a) at (0,0) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
101 \node[outer sep=-1pt] (x) at (0,0){}; |
181 \node[outer sep=-1pt] (x) at (0,0){}; |
146 %$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
226 %$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
147 by first computing the entire blob homology. |
227 by first computing the entire blob homology. |
148 \end{conj} |
228 \end{conj} |
149 \end{frame} |
229 \end{frame} |
150 |
230 |
151 \begin{frame}{$n$-categories} |
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152 \begin{block}{Defining $n$-categories is fraught with difficulties} |
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153 For now, 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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154 \end{block} |
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155 \begin{block}{} |
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156 Later, I'll explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. |
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157 \end{block} |
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158 |
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159 \begin{block}{} |
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160 \begin{itemize} |
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161 \item |
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162 Defining $n$-categories: a choice of `shape' for morphisms. |
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163 \item |
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164 We allow all shapes! A vector space for every ball. |
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165 \item |
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166 `Strong duality' is integral in our definition. |
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167 \end{itemize} |
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168 \end{block} |
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169 \end{frame} |
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170 |
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171 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} |
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172 |
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173 \section{Definition} |
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174 \begin{frame}{Fields and pasting diagrams} |
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175 \begin{block}{Pasting diagrams} |
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176 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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177 \end{block} |
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178 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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179 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
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180 \end{example} |
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181 \begin{block}{} |
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182 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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183 \vspace{-3mm} |
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184 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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185 \end{block} |
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186 \end{frame} |
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187 |
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188 \begin{frame}{Background: TQFT invariants} |
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189 \begin{defn} |
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190 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |
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191 \end{defn} |
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192 (`decapitated': no numerical invariants of $n+1$-manifolds.) |
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193 |
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194 \begin{block}{} |
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195 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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196 \end{block} |
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197 |
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198 \begin{block}{} |
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199 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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200 \end{block} |
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201 |
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202 \begin{defn} |
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203 Given an $n$-category $\cC$, the associated TQFT is |
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204 \vspace{-3mm} |
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205 $$\cA(\cM) = \cF(M) / \ker{ev},$$ |
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206 |
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207 \vspace{-3mm} |
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208 fields modulo fields which evaluate to zero inside some ball. |
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209 \end{defn} |
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210 \end{frame} |
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211 |
231 |
212 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
232 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
213 \begin{block}{Motivation} |
233 \begin{block}{Motivation} |
214 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. |
234 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM; \cC)$. |
215 \end{block} |
235 \end{block} |
216 |
236 |
217 \mode<handout>{\vspace{-5mm}} |
237 \mode<handout>{\vspace{-5mm}} |
218 \begin{block}{} |
238 \begin{block}{} |
219 \center |
239 \center |
220 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
240 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. |
221 \end{block} |
241 \end{block} |
222 |
242 |
223 \begin{block}{} |
243 \begin{block}{} |
224 \vspace{-1mm} |
244 \vspace{-1mm} |
225 $$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
245 $$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
304 \begin{block}{} |
324 \begin{block}{} |
305 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
325 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
306 \end{block} |
326 \end{block} |
307 \end{frame} |
327 \end{frame} |
308 |
328 |
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329 \mode<beamer>{ |
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330 \begin{frame}{An action of $\CH{\cM}$} |
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331 \begin{proof} |
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332 \begin{description} |
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333 \item[Step 1] If $\cM=B^n$ or a union of balls, there's a unique chain map, since $\bc_*(B^n; \cC) \htpy \cC$ is concentrated in homological degree $0$. |
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334 \item[Step 2] Fix an open cover $\cU$ of balls. \\ A family of homeomorphisms $P^k \to \Homeo(\cM)$ can be broken up in into pieces, each of which is supported in at most $k$ open sets from $\cU$. \qedhere |
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335 \end{description} |
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336 \end{proof} |
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337 \end{frame} |
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338 } |
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339 |
309 \begin{frame}{Gluing} |
340 \begin{frame}{Gluing} |
310 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
341 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
311 \begin{itemize} |
342 \begin{description} |
312 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
343 \item[multiplication ($m_2$):] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
313 \item[$m_k$:] reparametrising $[0,1]$ |
344 \item[associativity up to homotopy ($m_k$):] reparametrising $[0,1]$ using the action of $\CH{[0,1]}$. |
314 \end{itemize} |
345 \end{description} |
315 \end{block} |
346 \end{block} |
316 \begin{block}{} |
347 \begin{block}{} |
317 If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. |
348 If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. |
318 \end{block} |
349 \end{block} |
319 \begin{thm}[Gluing formula] |
350 \begin{thm}[Gluing formula] |
342 \end{block} |
373 \end{block} |
343 \end{frame} |
374 \end{frame} |
344 |
375 |
345 \begin{frame}{Maps to a space} |
376 \begin{frame}{Maps to a space} |
346 \begin{block}{} |
377 \begin{block}{} |
347 Fix a target space $T$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(T)$ defined by |
378 Fix a target space $\cT$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(\cT)$ defined by |
348 $$\pi_{\leq n}^\infty(T)(B) = C_*(\Maps(B\to T)).$$ |
379 $$\pi_{\leq n}^\infty(\cT)(B) = C_*(\Maps(B\to \cT)).$$ |
349 \end{block} |
380 \end{block} |
350 \begin{thm} |
381 \begin{thm} |
351 The blob complex recovers mapping spaces: |
382 The blob complex recovers mapping spaces: |
352 $$\bc_*(M; \pi_{\leq n}^\infty(T)) \iso C_*(\Maps(M \to T))$$ |
383 $$\bc_*(\cM; \pi_{\leq n}^\infty(\cT)) \iso C_*(\Maps(\cM \to \cT))$$ |
353 \end{thm} |
384 \end{thm} |
354 This generalizes a result of Lurie: if $T$ is $n-1$ connected, $\pi_{\leq n}^\infty(T)$ is an $E_n$-algebra and the blob complex is the same as his topological chiral homology. |
385 This generalizes a result of Lurie: if $\cT$ is $n-1$ connected, $\pi_{\leq n}^\infty(\cT)$ is an $E_n$-algebra and the blob complex is the same as his topological chiral homology. |
355 \end{frame} |
386 \end{frame} |
356 |
387 |
357 \end{document} |
388 \end{document} |
358 % ---------------------------------------------------------------- |
389 % ---------------------------------------------------------------- |
359 |
390 |