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\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}
\institute{UC Berkeley / Miller Institute for Basic Research}
\title{Blob homology, part $\mathbb{I}$}
\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}}

\begin{document}

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\section{Overview}

   \begin{frame}<beamer>
       \frametitle{Blob homology}
       \begin{quote}
      ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. 
        \end{quote}
        \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}}
      \tableofcontents
\end{frame}

\begin{frame}{What is \emph{blob homology}?}
\begin{block}{}
The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$.
\end{block}
\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]
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\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};
\uncover<1>{
\node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$};
\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};
\path[->](blobs) edge node[above] {$*= 0$} (skein);
}

\uncover<2>{
  \node[blue] (hoch) at (0,3) {$HH_*(\cC)$};
  \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};
  \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);
}

\uncover<3>{
  \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$};
  \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)};
  \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);
}

\end{tikzpicture}
\end{frame}

\begin{frame}{$n$-categories}
\begin{block}{Defining $n$-categories is fraught with difficulties}
I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories.
\end{block}
\begin{block}{}
Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.
\end{block}

\begin{block}{}
\begin{itemize}
\item
Defining $n$-categories: a choice of `shape' for morphisms.
\item
We allow all shapes! A vector space for every ball.
\item
`Strong duality' is integral in our definition.
\end{itemize}
\end{block}
\end{frame}

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

\section{Definition}
\begin{frame}{Fields and pasting diagrams}
\begin{block}{Pasting diagrams}
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 $\cC$.
\end{block}
\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]
$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$
\end{example}
\begin{block}{}
Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}.
\vspace{-3mm}
$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$
\end{block}
\end{frame}

\begin{frame}{Background: TQFT invariants}
\begin{defn}
A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$.
\end{defn}
(`decapitated': no numerical invariants of $n+1$-manifolds.)

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

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

\begin{defn}
Given an $n$-category $\cC$, the associated TQFT is 
\vspace{-3mm}
$$\cA(\cM) = \cF(M) / \ker{ev},$$

\vspace{-3mm}
fields modulo fields which evaluate to zero inside some ball.
\end{defn}
\end{frame}

\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}
\begin{block}{Motivation}
A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$.
\end{block}

\begin{block}{}
\center
$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$.
\end{block}

\begin{block}{}
\vspace{-1mm}
$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
\end{block}
\vspace{-3.5mm}
$$\mathfig{.5}{definition/single-blob}$$
\vspace{-3mm}
\begin{block}{}
\vspace{-6mm}
\begin{align*}
d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)
\end{align*}
\end{block}
\end{frame}

\begin{frame}{Definition, $k=2$}
\begin{block}{}
\vspace{-1mm}
$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$
\end{block}
\begin{block}{}
\vspace{-5mm}
\begin{align*}
\bc_2^{\text{disjoint}} & =  \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}
\end{align*}
\vspace{-4mm}
$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$
\end{block}
\begin{block}{}
\vspace{-5mm}
\begin{align*}
\bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}
\end{align*}
\vspace{-4mm}
$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$
\end{block}
\end{frame}

\begin{frame}{Definition, general case}
\begin{block}{}
$$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$
$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero.
\end{block}
\begin{block}{}
\vspace{-2mm}
$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$
\end{block}
\end{frame}

\section{Properties}
\begin{frame}{Hochschild homology}
\begin{block}{TQFT on $S^1$ is `coinvariants'}
\vspace{-3mm}
$$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$
\end{block}
\begin{block}{}
The Hochschild complex is `coinvariants of the bar resolution'
\vspace{-2mm}
$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$
\end{block}
\begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$]
$$m \tensor a \mapsto
\roundframe{\mathfig{0.35}{hochschild/1-chains}}
$$
\vspace{-5mm}
\begin{align*}
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} 
\end{align*}
\end{thm}
\end{frame}

\begin{frame}{An action of $\CH{\cM}$}
\begin{thm}
There's a chain map
$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$
which is associative up to homotopy, and compatible with gluing.
\end{thm}
\begin{block}{}
Taking $H_0$, this is the mapping class group acting on a TQFT skein module.
\end{block}
\end{frame}

\begin{frame}{Higher Deligne conjecture}
\begin{block}{Deligne conjecture}
Chains on the little discs operad acts on Hochschild cohomology.
\end{block}

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

\begin{block}{Theorem* (Higher Deligne conjecture)}
\scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.}
\vspace{-3mm}
$$\mathfig{.85}{deligne/manifolds}$$
\end{block}
\end{frame}

\begin{frame}{Gluing}
\begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category}
\begin{itemize}
\item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$
\item[$m_k$:] reparametrising $[0,1]$
\end{itemize}
\end{block}
\begin{block}{}
If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$.
\end{block}
\begin{thm}[Gluing formula]
When $Y \sqcup Y^{\text{op}} \subset \bdy X$,
\vspace{-5mm}
\[
	\bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow.
\]
\end{thm}
In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology.
\end{frame}
\end{document}
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