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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{The blob complex} |
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}} |
16 \date{Low-Dimensional Topology and Categorification, \\Stony Brook University, June 21-25 2010 \\ \begin{description}\item[slides:]\url{http://tqft.net/sunysb-blobs} \item[paper:]\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 |
22 \beamertemplatetransparentcovered |
22 \beamertemplatetransparentcovered |
23 |
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24 \setbeamertemplate{navigation symbols}{} % no navigation symbols, please |
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24 \mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}} |
27 \mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}} |
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26 \beamersetuncovermixins |
29 \beamersetuncovermixins |
27 {\opaqueness<1->{60}} |
30 {\opaqueness<1->{60}} |
30 |
33 |
31 |
34 |
32 \section{Overview} |
35 \section{Overview} |
33 |
36 |
34 \begin{frame}<beamer> |
37 \begin{frame}<beamer> |
35 \frametitle{Blob homology} |
38 \frametitle{The blob complex} |
36 \begin{quote} |
39 \begin{quote} |
37 ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. |
40 ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. |
38 \end{quote} |
41 \end{quote} |
39 \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} |
42 \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} |
40 \tableofcontents |
43 \tableofcontents |
41 \end{frame} |
44 \end{frame} |
42 |
45 |
43 \begin{frame}{What is \emph{blob homology}?} |
46 \begin{frame}{What is \emph{the blob complex}?} |
44 \begin{block}{} |
47 \begin{block}{} |
45 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
48 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$. |
46 \end{block} |
49 \end{block} |
47 \tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm] |
50 \tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm] |
48 \begin{tikzpicture}[] |
51 \begin{tikzpicture}[] |
51 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
54 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}}, |
52 again covered={\opaqueness<1->{50}} |
55 again covered={\opaqueness<1->{50}} |
53 } |
56 } |
54 |
57 |
55 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
58 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
56 \uncover<1>{ |
59 \uncover<2>{ |
57 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
60 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$}; |
58 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
61 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)}; |
59 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
62 \path[->](blobs) edge node[above] {$*= 0$} (skein); |
60 } |
63 } |
61 |
64 |
62 \uncover<2>{ |
65 \uncover<3>{ |
63 \node[blue] (hoch) at (0,3) {$HH_*(\cC)$}; |
66 \node[blue] (hoch) at (0,3) {$HH_*(\cC)$}; |
64 \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)}; |
67 \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)}; |
65 \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch); |
68 \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch); |
66 } |
69 } |
67 |
70 |
68 \uncover<3>{ |
71 \uncover<4>{ |
69 \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$}; |
72 \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$}; |
70 \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)}; |
73 \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)}; |
71 \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm); |
74 \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm); |
72 } |
75 } |
73 |
76 |
74 \end{tikzpicture} |
77 \end{tikzpicture} |
75 \end{frame} |
78 \end{frame} |
76 |
79 |
80 \begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} |
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81 \begin{thm} |
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82 Khovanov homology gives a $4$-category: |
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83 \begin{description} |
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84 \item[3-morphisms] tangles, with the usual $3$ operations, |
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85 \item[4-morphisms] $\Hom{Kh}{T_1}{T_2} = Kh(T_1 \cup \bar{T_2})$, composition defined by saddle cobordisms |
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86 \end{description} |
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87 \end{thm} |
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88 \begin{block}{} |
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89 There is a corresponding $4$-manifold invariant. Given $L \subset \bdy W^4$, it associates a doubly-graded vector space $\cA(W, L; Kh)$. |
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90 $$\cA(B^4, L; Kh) \iso Kh(L)$$ |
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91 \end{block} |
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92 \end{frame} |
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93 |
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94 \begin{frame}{Computations are hard} |
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95 \begin{block}{} |
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96 The corresponding $4$-manifold invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. |
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97 \vspace{-0.3cm} |
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98 \begin{align*} |
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99 \begin{tikzpicture} |
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100 \node(a) at (0,0) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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101 \node[outer sep=-1pt] (x) at (0,0){}; |
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102 \draw (x.45)-- (.5,.5); |
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103 \draw (x.135) -- (-.5,.5); |
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104 \draw (x.315) -- (.5,-.5); |
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105 \draw (x.45) -- (-.5,-.5); |
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106 \end{tikzpicture}\right)$}; |
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107 \node(b) at (-1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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108 \draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); |
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109 \draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); |
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110 \end{tikzpicture}\right)$}; |
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111 \node(c) at (1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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112 \draw (3.5,.5) .. controls (4,0) .. (4.5,.5); |
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113 \draw (3.5,-.5) .. controls (4,0) .. (4.5,-.5); |
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114 \end{tikzpicture}\right)$}; |
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115 \draw[->] (a) -- (b); |
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116 \draw[->] (b) -- (c); |
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117 \draw[->] (c) -- (a); |
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118 \end{tikzpicture} |
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119 \qquad \qquad |
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120 \begin{tikzpicture} |
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121 \node(a) at (0,0) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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122 \node[outer sep=-1pt] (x) at (0,0){}; |
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123 \draw (x.45)-- (.5,.5); |
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124 \draw (x.135) -- (-.5,.5); |
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125 \draw (x.315) -- (.5,-.5); |
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126 \draw (x.45) -- (-.5,-.5); |
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127 \end{tikzpicture}\right)$}; |
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128 \node(b) at (-1.4,-1.5) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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129 \draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); |
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130 \draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); |
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131 \end{tikzpicture}\right)$}; |
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132 \node(c) at (1.4,-1.5) {$\cA\left(M,\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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133 \draw (3.5,.5) .. controls (4,0) .. (4.5,.5); |
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134 \draw (3.5,-.5) .. controls (4,0) .. (4.5,-.5); |
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135 \end{tikzpicture}\right)$}; |
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136 \node at (0,-0.75) {\Large \color{red} ?}; |
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137 \draw[dashed] (a) -- (b); |
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138 \draw[dashed] (b) -- (c); |
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139 \draw[dashed] (c) -- (a); |
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140 \end{tikzpicture} |
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141 \end{align*}\vspace{-1cm} |
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142 \end{block} |
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143 There is a spectral sequence converging to $0$ relating the blob homologies for the triangle of resolutions. |
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144 \begin{conj} |
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145 It may be possible to compute the skein module |
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146 %$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
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147 by first computing the entire blob homology. |
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148 \end{conj} |
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149 \end{frame} |
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150 |
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77 \begin{frame}{$n$-categories} |
151 \begin{frame}{$n$-categories} |
78 \begin{block}{Defining $n$-categories is fraught with difficulties} |
152 \begin{block}{Defining $n$-categories is fraught with difficulties} |
79 I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. |
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. |
80 \end{block} |
154 \end{block} |
81 \begin{block}{} |
155 \begin{block}{} |
82 Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. |
156 Later, I'll explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. |
83 \end{block} |
157 \end{block} |
84 |
158 |
85 \begin{block}{} |
159 \begin{block}{} |
86 \begin{itemize} |
160 \begin{itemize} |
87 \item |
161 \item |
100 \begin{frame}{Fields and pasting diagrams} |
174 \begin{frame}{Fields and pasting diagrams} |
101 \begin{block}{Pasting diagrams} |
175 \begin{block}{Pasting diagrams} |
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 $\cC$. |
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 $\cC$. |
103 \end{block} |
177 \end{block} |
104 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
178 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
179 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
106 \end{example} |
180 \end{example} |
107 \begin{block}{} |
181 \begin{block}{} |
108 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
182 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
109 \vspace{-3mm} |
183 \vspace{-3mm} |
110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
184 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
138 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
212 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
139 \begin{block}{Motivation} |
213 \begin{block}{Motivation} |
140 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. |
214 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$. |
141 \end{block} |
215 \end{block} |
142 |
216 |
217 \mode<handout>{\vspace{-5mm}} |
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143 \begin{block}{} |
218 \begin{block}{} |
144 \center |
219 \center |
145 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
220 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
146 \end{block} |
221 \end{block} |
147 |
222 |
151 \end{block} |
226 \end{block} |
152 \vspace{-3.5mm} |
227 \vspace{-3.5mm} |
153 $$\mathfig{.5}{definition/single-blob}$$ |
228 $$\mathfig{.5}{definition/single-blob}$$ |
154 \vspace{-3mm} |
229 \vspace{-3mm} |
155 \begin{block}{} |
230 \begin{block}{} |
231 \mode<handout>{\vspace{-5mm}} |
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156 \vspace{-6mm} |
232 \vspace{-6mm} |
157 \begin{align*} |
233 \begin{align*} |
158 d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) |
234 d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) |
159 \end{align*} |
235 \end{align*} |
160 \end{block} |
236 \end{block} |
161 \end{frame} |
237 \end{frame} |
162 |
238 |
163 \begin{frame}{Definition, $k=2$} |
239 \begin{frame}{Definition, $k=2$} |
164 \begin{block}{} |
240 \begin{block}{} |
165 \vspace{-1mm} |
241 \vspace{-1mm} |
242 \mode<handout>{\vspace{-5mm}} |
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166 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
243 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
167 \end{block} |
244 \end{block} |
168 \begin{block}{} |
245 \begin{block}{} |
246 \mode<handout>{\vspace{-5mm}} |
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169 \vspace{-5mm} |
247 \vspace{-5mm} |
170 \begin{align*} |
248 \begin{align*} |
171 \bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
249 \bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
172 \end{align*} |
250 \end{align*} |
173 \vspace{-4mm} |
251 \vspace{-4mm} |
198 \begin{frame}{Hochschild homology} |
276 \begin{frame}{Hochschild homology} |
199 \begin{block}{TQFT on $S^1$ is `coinvariants'} |
277 \begin{block}{TQFT on $S^1$ is `coinvariants'} |
200 \vspace{-3mm} |
278 \vspace{-3mm} |
201 $$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$ |
279 $$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$ |
202 \end{block} |
280 \end{block} |
281 \mode<handout>{\vspace{-3mm}} |
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203 \begin{block}{} |
282 \begin{block}{} |
204 The Hochschild complex is `coinvariants of the bar resolution' |
283 The Hochschild complex is `coinvariants of the bar resolution' |
205 \vspace{-2mm} |
284 \vspace{-2mm} |
206 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
285 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
207 \end{block} |
286 \end{block} |
222 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
301 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
223 which is associative up to homotopy, and compatible with gluing. |
302 which is associative up to homotopy, and compatible with gluing. |
224 \end{thm} |
303 \end{thm} |
225 \begin{block}{} |
304 \begin{block}{} |
226 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
305 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
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} |
306 \end{block} |
244 \end{frame} |
307 \end{frame} |
245 |
308 |
246 \begin{frame}{Gluing} |
309 \begin{frame}{Gluing} |
247 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
310 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
260 \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. |
323 \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. |
261 \] |
324 \] |
262 \end{thm} |
325 \end{thm} |
263 In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. |
326 In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. |
264 \end{frame} |
327 \end{frame} |
328 |
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329 \begin{frame}{Higher Deligne conjecture} |
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330 \begin{block}{Deligne conjecture} |
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331 Chains on the little discs operad acts on Hochschild cohomology. |
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332 \end{block} |
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333 |
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334 \begin{block}{} |
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335 Call $\Hom{\bc_*(\bdy M)}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'. |
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336 \end{block} |
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337 |
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338 \begin{block}{Theorem (Higher Deligne conjecture)} |
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339 \scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.} |
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340 \vspace{-3mm} |
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341 $$\mathfig{.85}{deligne/manifolds}$$ |
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342 \end{block} |
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343 \end{frame} |
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344 |
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345 \begin{frame}{Maps to a space} |
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346 \begin{block}{} |
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347 Fix a target space $T$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(T)$ defined by |
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348 $$\pi_{\leq n}^\infty(T)(B) = C_*(\Maps(B\to T)).$$ |
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349 \end{block} |
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350 \begin{thm} |
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351 The blob complex recovers mapping spaces: |
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352 $$\bc_*(M; \pi_{\leq n}^\infty(T)) \iso C_*(\Maps(M \to T))$$ |
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353 \end{thm} |
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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. |
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355 \end{frame} |
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356 |
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265 \end{document} |
357 \end{document} |
266 % ---------------------------------------------------------------- |
358 % ---------------------------------------------------------------- |
267 |
359 |