author | Kevin Walker <kevin@canyon23.net> |
Wed, 18 Aug 2010 21:05:50 -0700 | |
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parent 394 | eac3c57c808a |
permissions | -rw-r--r-- |
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% use options |
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% '[beamer]' for a digital projector |
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% '[trans]' for an overhead projector |
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% '[handout]' for 4-up printed notes |
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\documentclass[beamer, compress]{beamer} |
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% change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. |
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\newcommand{\pathtotrunk}{../../} |
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\input{\pathtotrunk talks/talk_preamble.tex} |
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%\setbeameroption{previous slide on second screen=right} |
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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{The blob complex} |
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\date{ |
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Low-Dimensional Topology and Categorification, \\ |
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Stony Brook University, June 21-25 2010 \\ |
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\begin{description} |
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\item[slides:]\url{http://tqft.net/talks} |
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\item[paper:]\url{http://tqft.net/blobs} |
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% \item[shameless plug:]\url{http://mathoverflow.net} |
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\end{description} |
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} |
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\listfiles |
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\begin{document} |
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\frame{\titlepage} |
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\section{Overview} |
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\begin{frame}<beamer> |
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\frametitle{The blob complex} |
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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{the blob complex}?} |
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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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\uncover<2>{ |
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\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$}; |
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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<3>{ |
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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<4>{ |
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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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\section{TQFTs} |
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\begin{frame}{$n$-categories} |
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\begin{block}{There are many definitions of $n$-categories!} |
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For most of what follows, I'll draw $2$-dimensional pictures and rely on your intuition for pivotal categories. |
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\end{block} |
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\begin{block}{We have yet another definition: \emph{topological $n$-categories}} |
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\begin{itemize} |
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%\item A set $\cC(B^k)$ for every $k$-ball, $0 \leq k < n$. |
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\item A vector space $\cC(B^n)$ for every $n$-ball $B$. |
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%\item From these, inductively |
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%\begin{itemize} |
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%\item define a set $\cC(S^k)$ for each $k$-sphere, $0 \leq k < n$, |
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%\item require a map $\cC(B^k) \to \cC(S^{k-1})$. |
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%\end{itemize} |
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\item An associative gluing map: with $B = \bigcup_i B_i$, balls glued together to form a ball, |
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$$\bigotimes \cC(B_i) \to \cC(B)$$ |
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(the $\tensor$ is fibered over `boundary restriction' maps). |
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\item ... |
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\end{itemize} |
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\end{block} |
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These are easy to check for geometric examples, hard to check for algebraic examples. |
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\end{frame} |
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\begin{frame}{Cellulations of manifolds} |
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\begin{block}{} |
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Consider $\cell(M)$, the category of cellulations of a manifold $M$, with morphisms `antirefinements'. |
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\end{block} |
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\vspace{-4mm} |
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$$\mathfig{.35}{ncat/zz2}$$ |
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\vspace{-4mm} |
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\begin{block}{} |
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An $n$-category $\cC$ gives a functor from $\cell(M)$ to vector spaces. |
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\begin{description} |
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\item[objects] send a cellulation to the product of $\cC$ on each top-cell, restricting to the subset where boundaries agree |
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\item[morphisms] send an antirefinement to the appropriate gluing map. |
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\end{description} |
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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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\begin{frame}{Fields} |
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\begin{block}{} |
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A field on $\cM^n$ is a choice of cellulation and a choice of $n$-morphism for each top-cell (with matching boundaries). |
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%$$\cF(\cM) = \bigoplus_{\cX \in \cell(M)} \bigotimes_{B \in \cX} \cC(B)$$ |
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\end{block} |
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\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF\left(T^2\right)$$ |
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\end{example} |
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\begin{block}{} |
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Given a field on a ball, we can evaluate it to a morphism using the gluing map. 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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We associate to an $n$-manifold $\cM$ the skein module |
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\vspace{-1mm} |
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$$\cA(\cM) = \cF(\cM) / \ker{ev},\vspace{-1mm}$$ |
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fields modulo fields which evaluate to zero inside some ball. |
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\end{defn} |
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Equivalently, $\cA(\cM)$ is the colimit of $\cC$ along $\cell(M)$. |
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\vspace{4mm} |
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%\begin{itemize} |
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%\item We can also associate a $k$-category to an $n-k$-manifold. |
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%\item We don't assign a number to an $n+1$-manifold (a `decapitated' extended TQFT). |
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%\end{itemize} |
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$\cA(Y^{n-1} \times [0,1])$ is a $1$-category, and when $Y \subset \bdy X$, $\cA(X)$ is a module over $\cA(Y \times [0,1])$. |
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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{-1mm} |
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\[ |
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\cA(X \bigcup_Y \selfarrow) \iso \cA(X) \bigotimes_{\cA(Y \times [0,1])} \selfarrow. |
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\] |
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\end{thm} |
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\end{frame} |
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\begin{frame}{Motivation: Khovanov homology as a $4$d TQFT} |
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\begin{thm} |
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Khovanov homology gives a $4$-category: |
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\begin{description} |
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\item[3-morphisms] tangles, with the usual $3$ operations, |
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\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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\end{description} |
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\end{thm} |
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\begin{block}{} |
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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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$$\cA(B^4, L; Kh) \iso Kh(L)$$ |
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\end{block} |
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\end{frame} |
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|
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\begin{frame}{Computations are hard} |
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\begin{block}{} |
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This invariant is hard to compute, because the TQFT skein module construction breaks the exact triangle for resolving a crossing. |
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\vspace{-0.3cm} |
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\begin{align*} |
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\begin{tikzpicture} |
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\node(a) at (0,0) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\node[outer sep=-1pt] (x) at (0,0){}; |
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\draw (x.45)-- (.5,.5); |
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\draw (x.135) -- (-.5,.5); |
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\draw (x.315) -- (.5,-.5); |
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\draw (x.45) -- (-.5,-.5); |
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\end{tikzpicture}\right)$}; |
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\node(b) at (-1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); |
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\draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); |
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\end{tikzpicture}\right)$}; |
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\node(c) at (1.2,-1.5) {$Kh\left(\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\draw (3.5,.5) .. controls (4,0) .. (4.5,.5); |
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\draw (3.5,-.5) .. controls (4,0) .. (4.5,-.5); |
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\end{tikzpicture}\right)$}; |
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\draw[->] (a) -- (b); |
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\draw[->] (b) -- (c); |
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\draw[->] (c) -- (a); |
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\end{tikzpicture} |
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\qquad \qquad |
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\begin{tikzpicture} |
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\node(a) at (0,0) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\node[outer sep=-1pt] (x) at (0,0){}; |
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\draw (x.45)-- (.5,.5); |
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\end{tikzpicture}\right)$}; |
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\node(b) at (-1.4,-1.5) {$\cA\left(M, \begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\draw (1.5,.5) .. controls (2,0) .. (1.5,-.5); |
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\draw (2.5,.5) .. controls (2,0) .. (2.5,-.5); |
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\end{tikzpicture}\right)$}; |
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\node(c) at (1.4,-1.5) {$\cA\left(M,\begin{tikzpicture}[baseline=-2.5pt, scale=0.5, line width=1.5pt] |
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\end{tikzpicture}\right)$}; |
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\node at (0,-0.75) {\Large \color{red} ?}; |
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\draw[dashed] (a) -- (b); |
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\draw[dashed] (b) -- (c); |
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\draw[dashed] (c) -- (a); |
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\end{tikzpicture} |
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\end{align*}\vspace{-1cm} |
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\end{block} |
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There is a spectral sequence converging to $0$ relating the blob homologies for the triangle of resolutions. |
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\begin{conj} |
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It may be possible to compute the skein module |
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%$$\cA(W, L; Kh) = H_0(\bc_*(W, L; Kh))$$ |
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by first computing the entire blob homology. |
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\end{conj} |
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\end{frame} |
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|
388 | 230 |
\section{Definition} |
157 | 231 |
\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$} |
232 |
\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 $\cA(\cM; \cC)$. |
157 | 234 |
\end{block} |
235 |
||
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\mode<handout>{\vspace{-5mm}} |
157 | 237 |
\begin{block}{} |
238 |
\center |
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$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. |
157 | 240 |
\end{block} |
241 |
||
242 |
\begin{block}{} |
|
243 |
\vspace{-1mm} |
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$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
157 | 245 |
\end{block} |
246 |
\vspace{-3.5mm} |
|
247 |
$$\mathfig{.5}{definition/single-blob}$$ |
|
248 |
\vspace{-3mm} |
|
249 |
\begin{block}{} |
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\mode<handout>{\vspace{-5mm}} |
157 | 251 |
\vspace{-6mm} |
252 |
\begin{align*} |
|
253 |
d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC) |
|
254 |
\end{align*} |
|
255 |
\end{block} |
|
256 |
\end{frame} |
|
257 |
||
258 |
\begin{frame}{Definition, $k=2$} |
|
259 |
\begin{block}{} |
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\vspace{-1mm} |
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\mode<handout>{\vspace{-5mm}} |
157 | 262 |
$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
263 |
\end{block} |
|
264 |
\begin{block}{} |
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\mode<handout>{\vspace{-5mm}} |
157 | 266 |
\vspace{-5mm} |
267 |
\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} |
157 | 269 |
\end{align*} |
270 |
\vspace{-4mm} |
|
271 |
$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
|
272 |
\end{block} |
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\begin{block}{} |
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274 |
\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} |
157 | 277 |
\end{align*} |
278 |
\vspace{-4mm} |
|
279 |
$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
|
280 |
\end{block} |
|
153 | 281 |
\end{frame} |
282 |
||
170 | 283 |
\begin{frame}{Definition, general case} |
284 |
\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 fields that evaluate to zero. |
170 | 287 |
\end{block} |
288 |
\begin{block}{} |
|
289 |
\vspace{-2mm} |
|
290 |
$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
|
291 |
\end{block} |
|
292 |
\end{frame} |
|
293 |
||
294 |
\section{Properties} |
|
176 | 295 |
\begin{frame}{Hochschild homology} |
296 |
\begin{block}{TQFT on $S^1$ is `coinvariants'} |
|
297 |
\vspace{-3mm} |
|
388 | 298 |
$$\cA(S^1, A) = \Complex\set{\roundframe{ |
299 |
\tikz{\draw (0,0) circle (0.4); \foreach \q/\l in {90/a, 210/b, 330/c} {\draw[fill=red] (\q:0.4) circle (0.075); \node at (\q:0.6) {\l};}} |
|
300 |
}} |
|
301 |
\scalebox{2}{$/$} |
|
302 |
\set{\roundframe{\tikz{\draw (-30:0.4) arc (-30:210:0.4); \draw[fill=red] (90:0.4) circle (0.075); \node at (90:0.65) {$ab$};}} - \roundframe{ |
|
303 |
\tikz{\draw (-30:0.4) arc (-30:210:0.4); \foreach \q/\l in {120/a, 60/b} {\draw[fill=red] (\q:0.4) circle (0.075); \node at (\q:0.65) {\l};}}}} = A/(ab-ba)$$ |
|
176 | 304 |
\end{block} |
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\mode<handout>{\vspace{-3mm}} |
388 | 306 |
\begin{block}{Blob homology on $S^1$ is Hochschild homology} |
176 | 307 |
The Hochschild complex is `coinvariants of the bar resolution' |
308 |
\vspace{-2mm} |
|
309 |
$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$ |
|
388 | 310 |
|
311 |
We check universal properties, as it's hard to directly construct an isomorphism. |
|
312 |
\noop{ |
|
176 | 313 |
$$m \tensor a \mapsto |
314 |
\roundframe{\mathfig{0.35}{hochschild/1-chains}} |
|
315 |
$$ |
|
316 |
\vspace{-5mm} |
|
317 |
\begin{align*} |
|
318 |
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} |
|
319 |
\end{align*} |
|
388 | 320 |
} |
321 |
\end{block} |
|
176 | 322 |
\end{frame} |
323 |
||
170 | 324 |
\begin{frame}{An action of $\CH{\cM}$} |
325 |
\begin{thm} |
|
326 |
There's a chain map |
|
327 |
$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ |
|
328 |
which is associative up to homotopy, and compatible with gluing. |
|
329 |
\end{thm} |
|
330 |
\begin{block}{} |
|
331 |
Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
|
388 | 332 |
$$H_0(\Homeo(\cM)) \tensor \cA(\cM) \to \cA(\cM).$$ |
170 | 333 |
\end{block} |
334 |
\end{frame} |
|
335 |
||
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\mode<beamer>{ |
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\begin{frame}{An action of $\CH{\cM}$} |
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\begin{proof} |
394 | 339 |
Uniqueness: |
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\begin{description} |
394 | 341 |
\item[Step 1] If $\cM=B^n$ or a union of balls, there's a unique (up to homotopy) chain map, since $\bc_*(B^n; \cC) \htpy \cC$ is concentrated in homological degree $0$. |
342 |
\item[Step 2] Fix an open cover $\cU$ of balls. \\ A family of homeomorphisms $P^k \to \Homeo(\cM)$ can be broken up in into pieces, each of which is supported in at most $k$ open sets from $\cU$. \end{description} |
|
343 |
Existence: |
|
344 |
\begin{description} |
|
345 |
\item[Step 3] Show that all of the choices available above can be made consistently, using the method of acyclic models. \qedhere |
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\end{description} |
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\end{proof} |
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\end{frame} |
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} |
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|
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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{description} |
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\item[multiplication ($m_2$):] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
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\item[associativity up to homotopy ($m_k$):] reparametrising $[0,1]$ using the action of $\CH{[0,1]}$. |
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\end{description} |
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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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\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 $\Hom{\bc_*(\bdy M)}{\bc_*(\cM)}{\bc_*(\cM)}$ `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}{Maps to a space} |
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\begin{block}{} |
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Fix a target space $\cT$. There is an $A_\infty$ $n$-category $\pi_{\leq n}^\infty(\cT)$ defined by |
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$$\pi_{\leq n}^\infty(\cT)(B) = C_*(\Maps(B\to \cT)).$$ |
388 | 391 |
(Here $B$ is an $n$-ball.) |
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\end{block} |
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\begin{thm} |
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The blob complex recovers mapping spaces: |
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$$\bc_*(\cM; \pi_{\leq n}^\infty(\cT)) \iso C_*(\Maps(\cM \to \cT))$$ |
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\end{thm} |
388 | 397 |
This generalizes a result of Lurie: if $\cT$ is $n-1$ connected, $\pi_{\leq n}^\infty(\cT)$ is an $E_n$-algebra and in this special case the blob complex is presumably the same as his topological chiral homology. |
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\end{frame} |
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153 | 400 |
\end{document} |
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% ---------------------------------------------------------------- |
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