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\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}
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\institute{UC Berkeley / Miller Institute for Basic Research}
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\title{Blob homology, part $\mathbb{I}$}
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\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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\begin{document}
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\frame{\titlepage}
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{\opaqueness<1->{60}}
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{}
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\section{Overview}
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\begin{frame}<beamer>
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\frametitle{Blob homology}
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\begin{quote}
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... 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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\end{quote}
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\flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}}
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\tableofcontents
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\end{frame}
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\begin{frame}{What is \emph{blob homology}?}
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\begin{block}{}
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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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\end{block}
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\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]
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\begin{tikzpicture}[]
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\setbeamercovered{%
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transparent=5,
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% still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}},
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again covered={\opaqueness<1->{50}}
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}
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\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};
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\uncover<1>{
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\node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$};
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\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};
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\path[->](blobs) edge node[above] {$*= 0$} (skein);
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}
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\uncover<2>{
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\node[blue] (hoch) at (0,3) {$HH_*(\cC)$};
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\node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};
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\path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);
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}
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\uncover<3>{
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\node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$};
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\node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)};
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\path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);
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}
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\end{tikzpicture}
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\end{frame}
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\begin{frame}{$n$-categories}
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\begin{block}{Defining $n$-categories is fraught with difficulties}
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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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\end{block}
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\begin{block}{}
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Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.
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\end{block}
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\begin{block}{}
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\begin{itemize}
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\item
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Defining $n$-categories: a choice of `shape' for morphisms.
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\item
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We allow all shapes! A vector space for every ball.
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\item
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`Strong duality' is integral in our definition.
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\end{itemize}
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\end{block}
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\end{frame}
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\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}}
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\section{Definition}
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\begin{frame}{Fields and pasting diagrams}
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\begin{block}{Pasting diagrams}
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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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\end{block}
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\end{frame}
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\begin{frame}{Background: TQFT invariants}
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\begin{defn}
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A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$.
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\end{defn}
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(`decapitated': no numerical invariants of $n+1$-manifolds.)
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\begin{block}{}
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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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\end{block}
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\begin{block}{}
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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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\end{block}
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\begin{defn}
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Given an $n$-category $\cC$, the associated TQFT is
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\vspace{-3mm}
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$$\cA(\cM) = \cF(M) / \ker{ev},$$
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\vspace{-3mm}
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fields modulo fields which evaluate to zero inside some ball.
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\end{defn}
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\end{frame}
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\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}
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\begin{block}{Motivation}
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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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\end{block}
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\begin{block}{}
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\center
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$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$.
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\end{block}
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\begin{block}{}
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\vspace{-1mm}
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$$\bc_1(\cM; \cC) = \setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
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\end{block}
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\vspace{-3.5mm}
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$$\mathfig{.5}{definition/single-blob}$$
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\vspace{-3mm}
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\begin{block}{}
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\vspace{-6mm}
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\begin{align*}
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d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)
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\end{align*}
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\end{block}
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\end{frame}
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\begin{frame}{Definition, $k=2$}
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\begin{block}{}
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\vspace{-1mm}
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$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$
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\end{block}
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\begin{block}{}
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\vspace{-5mm}
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\begin{align*}
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\bc_2^{\text{disjoint}} & = \setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}
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\end{align*}
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\vspace{-4mm}
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$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$
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\end{block}
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\begin{block}{}
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\vspace{-5mm}
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\begin{align*}
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\bc_2^{\text{nested}} & = \setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}
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\end{align*}
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\vspace{-4mm}
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$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$
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\end{block}
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\end{frame}
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\begin{frame}{Definition, general case}
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\begin{block}{}
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$$\bc_k = \set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$
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$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero.
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\end{block}
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\begin{block}{}
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\vspace{-2mm}
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$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$
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\end{block}
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\end{frame}
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\section{Properties}
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\begin{frame}{Hochschild homology}
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\begin{block}{TQFT on $S^1$ is `coinvariants'}
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\vspace{-3mm}
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$$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$
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\end{block}
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\begin{block}{}
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The Hochschild complex is `coinvariants of the bar resolution'
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\vspace{-2mm}
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$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$
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\end{block}
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\begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$]
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$$m \tensor a \mapsto
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\roundframe{\mathfig{0.35}{hochschild/1-chains}}
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$$
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\vspace{-5mm}
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\begin{align*}
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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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\end{align*}
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\end{thm}
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\end{frame}
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\begin{frame}{An action of $\CH{\cM}$}
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\begin{thm}
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There's a chain map
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$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$
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which is associative up to homotopy, and compatible with gluing.
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\end{thm}
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\begin{block}{}
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Taking $H_0$, this is the mapping class group acting on a TQFT skein module.
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\end{block}
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\end{frame}
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\begin{frame}{Gluing}
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\begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category}
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\begin{itemize}
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\item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$
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\item[$m_k$:] reparametrising $[0,1]$
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\end{itemize}
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\end{block}
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\begin{block}{}
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If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$.
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\end{block}
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\begin{thm}[Gluing formula]
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When $Y \sqcup Y^{\text{op}} \subset \bdy X$,
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\vspace{-5mm}
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\[
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\bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow.
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\]
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\end{thm}
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In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology.
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\end{frame}
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\end{document}
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% ----------------------------------------------------------------
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