# HG changeset patch # User Scott Morrison # Date 1288037295 25200 # Node ID e0f5ec5827254a6fb45b02f7dcfe2270269dd2c5 # Parent f958e0ea62f83cd1c64df447ac8e1ed877fe1ea0 incorporating statements of results in PNAS article diff -r f958e0ea62f8 -r e0f5ec582725 blob1.tex --- a/blob1.tex Sun Oct 24 22:48:18 2010 -0700 +++ b/blob1.tex Mon Oct 25 13:08:15 2010 -0700 @@ -81,7 +81,7 @@ % ---------------------------------------------------------------- %\newcommand{\urlprefix}{} -\bibliographystyle{plain} +\bibliographystyle{alpha} \bibliography{bibliography/bibliography} % ---------------------------------------------------------------- diff -r f958e0ea62f8 -r e0f5ec582725 pnas/pnas.tex --- a/pnas/pnas.tex Sun Oct 24 22:48:18 2010 -0700 +++ b/pnas/pnas.tex Mon Oct 25 13:08:15 2010 -0700 @@ -74,7 +74,6 @@ %\def\s{\sigma} \input{preamble} -%\input{../text/article_preamble} \input{../text/kw_macros} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -161,7 +160,249 @@ %% \subsubsection{} -\input{../text/intro} +\section{} + +\nn{ +background: TQFTs are important, historically, semisimple categories well-understood. +Many new examples arising recently which do not fit this framework, e.g. SW and OS theory. +These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); +it would be nice to give generalized TQFT axioms that encompass these. +Triangulated categories are important; often calculations are via exact sequences, +and the standard TQFT constructions are quotients, which destroy exactness. +A first attempt to deal with this might be to replace all the tensor products in gluing formulas +with derived tensor products (cite Kh?). +However, in this approach it's probably difficult to prove invariance of constructions, +because they depend on explicit presentations of the manifold. +We'll give a manifestly invariant construction, +and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.} + +\section{Definitions} +\subsection{$n$-categories} +\nn{ +Axioms for $n$-categories, examples (maps, string diagrams) +} +\nn{ +Decide if we need a friendlier, skein-module version. +} +\subsection{The blob complex} +\subsubsection{Decompositions of manifolds} +\nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls} +\subsubsection{Homotopy colimits} +\nn{How can we extend an $n$-category from balls to arbitrary manifolds?} + +\nn{In practice, this gives the old definition} +\subsubsection{} +\section{Properties of the blob complex} +\subsection{Formal properties} +\label{sec:properties} +The blob complex enjoys the following list of formal properties. + +\begin{property}[Functoriality] +\label{property:functoriality}% +The blob complex is functorial with respect to homeomorphisms. +That is, +for a fixed $n$-dimensional system of fields $\cF$, the association +\begin{equation*} +X \mapsto \bc_*(X; \cF) +\end{equation*} +is a functor from $n$-manifolds and homeomorphisms between them to chain +complexes and isomorphisms between them. +\end{property} +As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; +this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. + +\begin{property}[Disjoint union] +\label{property:disjoint-union} +The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. +\begin{equation*} +\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) +\end{equation*} +\end{property} + +If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, +write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. +Note that this includes the case of gluing two disjoint manifolds together. +\begin{property}[Gluing map] +\label{property:gluing-map}% +%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map +%\begin{equation*} +%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). +%\end{equation*} +Given a gluing $X \to X_\mathrm{gl}$, there is +a natural map +\[ + \bc_*(X) \to \bc_*(X_\mathrm{gl}) +\] +(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). +\end{property} + +\begin{property}[Contractibility] +\label{property:contractibility}% +With field coefficients, the blob complex on an $n$-ball is contractible in the sense +that it is homotopic to its $0$-th homology. +Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces +associated by the system of fields $\cF$ to balls. +\begin{equation*} +\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} +\end{equation*} +\end{property} + +\nn{Properties \ref{property:functoriality} will be immediate from the definition given in +\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. +Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and +\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} + +\subsection{Specializations} +\label{sec:specializations} + +The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. + +\begin{thm}[Skein modules] +\label{thm:skein-modules} +The $0$-th blob homology of $X$ is the usual +(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ +by $\cF$. +\begin{equation*} +H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) +\end{equation*} +\end{thm} + +\begin{thm}[Hochschild homology when $X=S^1$] +\label{thm:hochschild} +The blob complex for a $1$-category $\cC$ on the circle is +quasi-isomorphic to the Hochschild complex. +\begin{equation*} +\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} +\end{equation*} +\end{thm} + +Proposition \ref{thm:skein-modules} is immediate from the definition, and +Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. + + +\subsection{Structure of the blob complex} +\label{sec:structure} + +In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. + +\begin{thm}[$C_*(\Homeo(-))$ action] +\label{thm:CH}\label{thm:evaluation} +There is a chain map +\begin{equation*} +e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). +\end{equation*} +such that +\begin{enumerate} +\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. + +\item For +any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram +(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). +\begin{equation*} +\xymatrix@C+0.3cm{ + \CH{X} \otimes \bc_*(X) + \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & + \bc_*(X) \ar[d]_{\gl_Y} \\ + \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) +} +\end{equation*} +\end{enumerate} + +Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). +\begin{equation*} +\xymatrix{ +\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ +\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) +} +\end{equation*} +\end{thm} + +Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps +$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ +for any homeomorphic pair $X$ and $Y$, +satisfying corresponding conditions. + +\begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] +\label{thm:blobs-ainfty} +Let $\cC$ be a topological $n$-category. +Let $Y$ be an $n{-}k$-manifold. +There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, +to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set +$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ +(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) +These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in +Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. +\end{thm} +\begin{rem} +Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. +We think of this $A_\infty$ $n$-category as a free resolution. +\end{rem} +This result is described in more detail as Example 6.2.8 of \cite{1009.5025} + +The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. +%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. + +\newtheorem*{thm:product}{Theorem \ref{thm:product}} + +\begin{thm}[Product formula] +\label{thm:product} +Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. +Let $\cC$ be an $n$-category. +Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). +Then +\[ + \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). +\] +\end{thm} +The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps +(see \cite[\S7.1]{1009.5025}). + +Fix a topological $n$-category $\cC$, which we'll omit from the notation. +Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. + +\begin{thm}[Gluing formula] +\label{thm:gluing} +\mbox{}% <-- gets the indenting right +\begin{itemize} +\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an +$A_\infty$ module for $\bc_*(Y)$. + +\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of +$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: +\begin{equation*} +\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow +\end{equation*} +\end{itemize} +\end{thm} + +\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} + +\section{Applications} +\label{sec:applications} +Finally, we give two applications of the above machinery. + +\begin{thm}[Mapping spaces] +\label{thm:map-recon} +Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps +$B^n \to T$. +(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) +Then +$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ +\end{thm} + +This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. +Note that there is no restriction on the connectivity of $T$ as in \cite[Theorem 3.8.6]{0911.0018}. +\nn{The proof appears in \S \ref{sec:map-recon}.} + + +\begin{thm}[Higher dimensional Deligne conjecture] +\label{thm:deligne} +The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. +Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, +this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. +\end{thm} +\nn{See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.} + %% == end of paper: @@ -209,9 +450,15 @@ %% Enter the largest bibliography number in the facing curly brackets %% following \begin{thebibliography} -\begin{thebibliography}{} +%%%% BIBTEX +\bibliographystyle{alpha} +\bibliography{../bibliography/bibliography} -\end{thebibliography} +%%%% non-BIBTEX +%\begin{thebibliography}{} +% +%\end{thebibliography} + \end{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff -r f958e0ea62f8 -r e0f5ec582725 pnas/preamble.tex --- a/pnas/preamble.tex Sun Oct 24 22:48:18 2010 -0700 +++ b/pnas/preamble.tex Mon Oct 25 13:08:15 2010 -0700 @@ -9,7 +9,7 @@ \newcommand{\CM}[2]{C_*(\Maps(#1 \to #2))} \newcommand{\CD}[1]{C_*(\Diff(#1))} -\newcommand{\CH}[1]{C_*(\Homeo(#1))} +\newcommand{\CH}[1]{CH_*(#1)} \newcommand{\cl}[1]{\underrightarrow{#1}} @@ -37,7 +37,7 @@ \newcommand{\tensor}{\otimes} \newcommand{\Tensor}{\bigotimes} -\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\draw[->] (0,0.2) .. controls (0.6,0.8) and (0.6,-0.6) .. (0,0);}}}} +\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.48,-0.16); \draw[->] (0,0.2) .. controls (0.6,0.8) and (0.6,-0.6) .. (0,0);}}}} \newcommand{\bdy}{\partial} \newcommand{\compose}{\circ} @@ -46,6 +46,9 @@ \newcommand{\id}{\boldsymbol{1}} \newtheorem{property}{Property} +\newtheorem{prop}{Proposition} +\newtheorem{thm}[prop]{Theorem} + \newenvironment{rem}{\noindent\textsl{Remark.}}{} % \mathrlap -- a horizontal \smash-------------------------------- @@ -63,6 +66,16 @@ \def\mathclapinternal#1#2{% \clap{$\mathsurround=0pt#1{#2}$}} +% references + +\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} +\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} +\newcommand{\euclid}[1]{\href{http://projecteuclid.org/euclid.cmp/#1}{{\tt at Project Euclid: #1}}} +\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} +\newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})} + + + % packages \usepackage{tikz} @@ -73,3 +86,16 @@ \usepackage[all,color]{xy} \SelectTips{cm}{} + +\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} + +\usepackage{xcolor} +\definecolor{dark-red}{rgb}{0.7,0.25,0.25} +\definecolor{dark-blue}{rgb}{0.15,0.15,0.55} +\definecolor{medium-blue}{rgb}{0,0,0.65} + +\hypersetup{ + colorlinks, linkcolor={dark-red}, + citecolor={dark-blue}, urlcolor={medium-blue} +} + diff -r f958e0ea62f8 -r e0f5ec582725 preamble.tex --- a/preamble.tex Sun Oct 24 22:48:18 2010 -0700 +++ b/preamble.tex Mon Oct 25 13:08:15 2010 -0700 @@ -160,7 +160,7 @@ %\newsavebox{\selfarrowcontents} %\savebox{\selfarrowcontents}{\selfarrow} %\renewcommand{\selfarrow}{\usebox{\selfarrowcontents}} -\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\draw[->] (0,0.2) .. controls (0.6,0.8) and (0.6,-0.6) .. (0,0);}}}} +\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.48,-0.16); \draw[->] (0,0.2) .. controls (0.6,0.8) and (0.6,-0.6) .. (0,0);}}}} \newcommand{\CM}[2]{C_*(\Maps(#1 \to #2))} \newcommand{\CD}[1]{C_*(\Diff(#1))}