
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.
 




