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     3 %  '[trans]' for an overhead projector

     4 %  '[handout]' for 4-up printed notes

     5 \documentclass[beamer, compress]{beamer}

     6 

     7 % change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes.

     8 \newcommand{\pathtotrunk}{../../}

     9 \input{\pathtotrunk talks/talk_preamble.tex}

    10 

    11 %\setbeameroption{previous slide on second screen=right}

    12 

    13 \author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}

    14 \institute{UC Berkeley / Miller Institute for Basic Research}

    15 \title{Blob homology, part $\mathbb{I}$}

    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 

    18 \begin{document}

    19 

    20 \frame{\titlepage}

    21 

    22 \beamertemplatetransparentcovered 

    23 

    24 \mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}

    25 

    26 \beamersetuncovermixins 

    27 {\opaqueness<1->{60}} 

    28 {} 

    29 

    30 

    31 

    32 \section{Overview}

    33 

    34    \begin{frame}<beamer>

    35        \frametitle{Blob homology}

    36        \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. 

    38         \end{quote}

    39         \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}}

    40       \tableofcontents

    41 \end{frame}

    42 

    43 \begin{frame}{What is \emph{blob homology}?}

    44 \begin{block}{}

    45 The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$.

    46 \end{block}

    47 \tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]

    48 \begin{tikzpicture}[]

    49 \setbeamercovered{%

    50  transparent=5,

    51 % still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}},

    52  again covered={\opaqueness<1->{50}}

    53 }

    54 

    55 \node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};

    56 \uncover<1>{

    57 \node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$};

    58 \node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};

    59 \path[->](blobs) edge node[above] {$*= 0$} (skein);

    60 }

    61 

    62 \uncover<2>{

    63   \node[blue] (hoch) at (0,3) {$HH_*(\cC)$};

    64   \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};

    65   \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);

    66 }

    67 

    68 \uncover<3>{

    69   \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)};

    71   \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);

    72 }

    73 

    74 \end{tikzpicture}

    75 \end{frame}

    76 

    77 \begin{frame}{$n$-categories}

    78 \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.

    80 \end{block}

    81 \begin{block}{}

    82 Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.

    83 \end{block}

    84 

    85 \begin{block}{}

    86 \begin{itemize}

    87 \item

    88 Defining $n$-categories: a choice of `shape' for morphisms.

    89 \item

    90 We allow all shapes! A vector space for every ball.

    91 \item

    92 `Strong duality' is integral in our definition.

    93 \end{itemize}

    94 \end{block}

    95 \end{frame}

    96 

    97 \newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}}

    98 

    99 \section{Definition}

   100 \begin{frame}{Fields and pasting diagrams}

   101 \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 $\cC$.

   103 \end{block}

   104 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]

   105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$

   106 \end{example}

   107 \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}.

   109 \vspace{-3mm}

   110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$

   111 \end{block}

   112 \end{frame}

   113 

   114 \begin{frame}{Background: TQFT invariants}

   115 \begin{defn}

   116 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$.

   117 \end{defn}

   118 (`decapitated': no numerical invariants of $n+1$-manifolds.)

   119 

   120 \begin{block}{}

   121 If the manifold has boundary, we get a category. Objects are boundary data, $\Hom{\cA(\cM)}{x}{y} = \cA(\cM; x,y)$.

   122 \end{block}

   123 

   124 \begin{block}{}

   125 We want to extend `all the way down'. The $k$-category associated to the $n-k$-manifold $\cY$ is $\cA(\cY \times B^k)$.

   126 \end{block}

   127 

   128 \begin{defn}

   129 Given an $n$-category $\cC$, the associated TQFT is 

   130 \vspace{-3mm}

   131 $$\cA(\cM) = \cF(M) / \ker{ev},$$

   132 

   133 \vspace{-3mm}

   134 fields modulo fields which evaluate to zero inside some ball.

   135 \end{defn}

   136 \end{frame}

   137 

   138 \begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}

   139 \begin{block}{Motivation}

   140 A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$.

   141 \end{block}

   142 

   143 \begin{block}{}

   144 \center

   145 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$.

   146 \end{block}

   147 

   148 \begin{block}{}

   149 \vspace{-1mm}

   150 $$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$

   151 \end{block}

   152 \vspace{-3.5mm}

   153 $$\mathfig{.5}{definition/single-blob}$$

   154 \vspace{-3mm}

   155 \begin{block}{}

   156 \vspace{-6mm}

   157 \begin{align*}

   158 d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)

   159 \end{align*}

   160 \end{block}

   161 \end{frame}

   162 

   163 \begin{frame}{Definition, $k=2$}

   164 \begin{block}{}

   165 \vspace{-1mm}

   166 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$

   167 \end{block}

   168 \begin{block}{}

   169 \vspace{-5mm}

   170 \begin{align*}

   171 \bc_2^{\text{disjoint}} & =  \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}

   172 \end{align*}

   173 \vspace{-4mm}

   174 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$

   175 \end{block}

   176 \begin{block}{}

   177 \vspace{-5mm}

   178 \begin{align*}

   179 \bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}

   180 \end{align*}

   181 \vspace{-4mm}

   182 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$

   183 \end{block}

   184 \end{frame}

   185 

   186 \begin{frame}{Definition, general case}

   187 \begin{block}{}

   188 $$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$

   189 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero.

   190 \end{block}

   191 \begin{block}{}

   192 \vspace{-2mm}

   193 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$

   194 \end{block}

   195 \end{frame}

   196 

   197 \section{Properties}

   198 \begin{frame}{Hochschild homology}

   199 \begin{block}{TQFT on $S^1$ is `coinvariants'}

   200 \vspace{-3mm}

   201 $$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$

   202 \end{block}

   203 \begin{block}{}

   204 The Hochschild complex is `coinvariants of the bar resolution'

   205 \vspace{-2mm}

   206 $$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$

   207 \end{block}

   208 \begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$]

   209 $$m \tensor a \mapsto

   210 \roundframe{\mathfig{0.35}{hochschild/1-chains}}

   211 $$

   212 \vspace{-5mm}

   213 \begin{align*}

   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} 

   215 \end{align*}

   216 \end{thm}

   217 \end{frame}

   218 

   219 \begin{frame}{An action of $\CH{\cM}$}

   220 \begin{thm}

   221 There's a chain map

   222 $$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$

   223 which is associative up to homotopy, and compatible with gluing.

   224 \end{thm}

   225 \begin{block}{}

   226 Taking $H_0$, this is the mapping class group acting on a TQFT skein module.

   227 \end{block}

   228 \end{frame}

   229 

   230 \begin{frame}{Higher Deligne conjecture}

   231 \begin{block}{Deligne conjecture}

   232 Chains on the little discs operad acts on Hochschild cohomology.

   233 \end{block}

   234 

   235 \begin{block}{}

   236 Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'.

   237 \end{block}

   238 

   239 \begin{block}{Theorem* (Higher Deligne conjecture)}

   240 \scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.}

   241 \vspace{-3mm}

   242 $$\mathfig{.85}{deligne/manifolds}$$

   243 \end{block}

   244 \end{frame}

   245 

   246 \begin{frame}{Gluing}

   247 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category}

   248 \begin{itemize}

   249 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$

   250 \item[$m_k$:] reparametrising $[0,1]$

   251 \end{itemize}

   252 \end{block}

   253 \begin{block}{}

   254 If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$.

   255 \end{block}

   256 \begin{thm}[Gluing formula]

   257 When $Y \sqcup Y^{\text{op}} \subset \bdy X$,

   258 \vspace{-5mm}

   259 \[

   260 	\bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow.

   261 \]

   262 \end{thm}

   263 In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology.

   264 \end{frame}

   265 \end{document}

   266 % ----------------------------------------------------------------

   267