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--- 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 @@
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-%\input{../text/article_preamble}
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 %% \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.}
+
 
 
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+%%%% BIBTEX
+\bibliographystyle{alpha}
+\bibliography{../bibliography/bibliography}
 
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+%%%% non-BIBTEX
+%\begin{thebibliography}{}
+%
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+
 
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