diff -r 979fbe9a14e8 -r a02a6158f3bd text/intro.tex --- a/text/intro.tex Fri Jun 25 09:48:24 2010 -0700 +++ b/text/intro.tex Sat Jun 26 16:31:28 2010 -0700 @@ -3,13 +3,13 @@ \section{Introduction} We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. -This blob complex provides a simultaneous generalisation of several well-understood constructions: +This blob complex provides a simultaneous generalization of several well-understood constructions: \begin{itemize} \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. -(See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) +(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. -(See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) +(See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains on the configuration space of unlabeled points in $M$. @@ -23,16 +23,16 @@ The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, -extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing -formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). +extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing +formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}). We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail. \subsubsection{Structure of the paper} -The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), -summarise the formal properties of the blob complex (see \S \ref{sec:properties}) -and outline anticipated future directions and applications (see \S \ref{sec:future}). +The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), +summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) +and outline anticipated future directions (see \S \ref{sec:future}). The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. @@ -64,7 +64,7 @@ Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), -in particular the `gluing formula' of Property \ref{property:gluing} below. +in particular the `gluing formula' of Theorem \ref{thm:gluing} below. The relationship between all these ideas is sketched in Figure \ref{fig:outline}. @@ -101,7 +101,7 @@ Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ } (FU); -\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Property \ref{property:skein-modules}} (A); +\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); @@ -217,7 +217,7 @@ complexes and isomorphisms between them. \end{property} As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; -this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. +this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here. @@ -256,8 +256,17 @@ \end{equation} \end{property} -\begin{property}[Skein modules] -\label{property:skein-modules}% +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 $\cC$. @@ -265,23 +274,30 @@ \begin{equation*} H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) \end{equation*} -\end{property} +\end{thm} -\todo{Somehow, the Hochschild homology thing isn't a "property". -Let's move it and call it a theorem? -S} -\begin{property}[Hochschild homology when $X=S^1$] -\label{property:hochschild}% +\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} + +\begin{thm:hochschild}[Hochschild homology when $X=S^1$] The blob complex for a $1$-category $\cC$ on the circle is quasi-isomorphic to the Hochschild complex. \begin{equation*} \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} \end{equation*} -\end{property} +\end{thm:hochschild} + +Theorem \ref{thm:skein-modules} is immediate from the definition, and +Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. +We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. + + +\subsection{Structure of the blob complex} +\label{sec:structure} In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. -\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\ +\begin{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\ \vspace{-0.5cm} -\label{property:evaluation}% +\label{thm:evaluation}% \begin{enumerate} \item There is a chain map \begin{equation*} @@ -311,7 +327,7 @@ } \end{equation*} \end{enumerate} -\end{property} +\end{thm} Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ @@ -322,8 +338,8 @@ Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. -\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] -\label{property:blobs-ainfty} +\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 $0 \leq m < k$, @@ -331,8 +347,8 @@ $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ (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 Property \ref{property:evaluation}. -\end{property} +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. @@ -342,24 +358,26 @@ instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. -\begin{property}[Product formula] -\label{property:product} +\newtheorem*{thm:product}{Theorem \ref{thm:product}} + +\begin{thm:product}[Product formula] 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 Property \ref{property:blobs-ainfty}). +Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}). Then \[ \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). \] -\end{property} +\end{thm:product} We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. 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. (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) -\begin{property}[Gluing formula] -\label{property:gluing}% +\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} + +\begin{thm:gluing}[Gluing formula] \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 @@ -371,32 +389,37 @@ \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow \end{equation*} \end{itemize} -\end{property} +\end{thm:gluing} + +Theorem \ref{thm:evaluation} is proved in +in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, +and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. -Finally, we prove two theorems which we consider as applications. +\subsection{Applications} +\label{sec:applications} +Finally, we give two theorems which we consider as applications. -\begin{thm}[Mapping spaces] +\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} + +\begin{thm:map-recon}[Mapping spaces] 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} +\end{thm:map-recon} -This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. +This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}. -\begin{thm}[Higher dimensional Deligne conjecture] -\label{thm:deligne} +\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} + +\begin{thm:deligne}[Higher dimensional Deligne conjecture] The singular chains of the $n$-dimensional fat graph operad act on blob cochains. -\end{thm} +\end{thm:deligne} See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. -Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in -\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. -Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. -Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} -in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, -and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. + + \subsection{Future directions} \label{sec:future} @@ -425,6 +448,6 @@ \subsection{Thanks and acknowledgements} We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, -Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. +Michael Freedman, Vaughan Jones, Alexander Kirillov, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley.