author | Scott Morrison <scott@tqft.net> |
Tue, 30 Nov 2010 11:33:26 -0800 | |
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parent 378 | e5867a64cae5 |
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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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\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
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\end{example} |
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\begin{block}{} |
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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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\vspace{-3mm} |
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$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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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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$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
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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}} & = \Complex\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}} & = \Complex\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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$$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ |
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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}{Higher Deligne conjecture} |
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\begin{block}{Deligne conjecture} |
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Chains on the little discs operad acts on Hochschild cohomology. |
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\end{block} |
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\begin{block}{} |
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Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'. |
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\end{block} |
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\begin{block}{Theorem* (Higher Deligne conjecture)} |
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\scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.} |
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\vspace{-3mm} |
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$$\mathfig{.85}{deligne/manifolds}$$ |
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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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