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13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} |
13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker} |
14 \institute{UC Berkeley / Miller Institute for Basic Research} |
14 \institute{UC Berkeley / Miller Institute for Basic Research} |
15 \title{Blob homology, part $\mathbb{I}$} |
15 \title{Blob homology, part $\mathbb{I}$} |
16 \date{Homotopy Theory and Higher Algebraic Structures, UC Riverside, November 10 2009 \\ \url{http://tqft.net/UCR-blobs1}} |
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}} |
17 |
17 |
18 \begin{document} |
18 \begin{document} |
19 |
19 |
20 \frame{\titlepage} |
20 \frame{\titlepage} |
21 |
21 |
52 again covered={\opaqueness<1->{50}} |
52 again covered={\opaqueness<1->{50}} |
53 } |
53 } |
54 |
54 |
55 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
55 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
56 \uncover<1>{ |
56 \uncover<1>{ |
57 \node[blue] (skein) at (4,0) {$A(\cM; \cC)$}; |
57 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
58 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
58 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
59 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
59 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
60 } |
60 } |
61 |
61 |
62 \uncover<2>{ |
62 \uncover<2>{ |
109 \vspace{-3mm} |
109 \vspace{-3mm} |
110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
111 \end{block} |
111 \end{block} |
112 \end{frame} |
112 \end{frame} |
113 |
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 |
114 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
138 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
115 \begin{block}{Motivation} |
139 \begin{block}{Motivation} |
116 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $A(\cM,; \cC)$. |
140 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. |
117 \end{block} |
141 \end{block} |
118 |
142 |
119 \begin{block}{} |
143 \begin{block}{} |
120 \center |
144 \center |
121 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
145 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
159 \end{block} |
183 \end{block} |
160 \end{frame} |
184 \end{frame} |
161 |
185 |
162 \begin{frame}{Definition, general case} |
186 \begin{frame}{Definition, general case} |
163 \begin{block}{} |
187 \begin{block}{} |
164 $$\bc_k = \set{\mathfig{0.4}{tempkw/blobkdiagram}}$$ |
188 $$\bc_k = \set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ |
165 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
189 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
166 \end{block} |
190 \end{block} |
167 \begin{block}{} |
191 \begin{block}{} |
168 \vspace{-2mm} |
192 \vspace{-2mm} |
169 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
193 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
170 \end{block} |
194 \end{block} |
171 \end{frame} |
195 \end{frame} |
172 |
196 |
173 \section{Properties} |
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 |
174 \begin{frame}{An action of $\CH{\cM}$} |
219 \begin{frame}{An action of $\CH{\cM}$} |
175 \begin{thm} |
220 \begin{thm} |
176 There's a chain map |
221 There's a chain map |
177 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
222 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
178 which is associative up to homotopy, and compatible with gluing. |
223 which is associative up to homotopy, and compatible with gluing. |