# HG changeset patch # User Scott Morrison # Date 1288040493 25200 # Node ID e5ab1b074d88800d3e0fecf4cbafe0c8ee53ea79 # Parent 8378e03d3c7fdf8513b3e07bce9229b805a00a70 minor edits and cleanup diff -r 8378e03d3c7f -r e5ab1b074d88 pnas/pnas.tex --- a/pnas/pnas.tex Mon Oct 25 13:36:12 2010 -0700 +++ b/pnas/pnas.tex Mon Oct 25 14:01:33 2010 -0700 @@ -187,18 +187,17 @@ \subsection{The blob complex} \subsubsection{Decompositions of manifolds} -A {\emph ball decomposition} of $W$ is a +A \emph{ball decomposition} of $W$ is a sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls -$\du_a X_a$. - -If $X_a$ is some component of $M_0$, note that its image in $W$ need not be a ball; parts of $\bd X_a$ may have been glued together. -Define a {\it permissible decomposition} of $W$ to be a map +$\du_a X_a$ and each $M_i$ is a manifold. +If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. +A {\it permissible decomposition} of $W$ is a map \[ \coprod_a X_a \to W, \] which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. -Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls -are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. +A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls +are glued up to yield $W$, and just require that there is some non-pathological way to do this. Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ @@ -215,16 +214,16 @@ a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets (possibly with additional structure if $k=n$). Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, -and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries +and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries are splittable along this decomposition. \begin{defn} Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset -\begin{equation} -\label{eq:psi-C} +\begin{equation*} +%\label{eq:psi-C} \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl -\end{equation} +\end{equation*} where the restrictions to the various pieces of shared boundaries amongst the cells $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. @@ -246,14 +245,14 @@ \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 +for a fixed $n$-category $\cC$, the association \begin{equation*} -X \mapsto \bc_*(X; \cF) +X \mapsto \bc_*(X; \cC) \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 $\bc_*(X; \cF)$; +As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. \begin{property}[Disjoint union] @@ -264,9 +263,8 @@ \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. +If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, +write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. \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 @@ -274,11 +272,11 @@ %\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 +a map \[ - \bc_*(X) \to \bc_*(X_\mathrm{gl}) + \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), \] -(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). +natural with respect to homeomorphisms, and associative with respect to iterated gluings. \end{property} \begin{property}[Contractibility] @@ -300,7 +298,7 @@ \subsection{Specializations} \label{sec:specializations} -The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. +The blob complex has two important special cases. \begin{thm}[Skein modules] \label{thm:skein-modules} @@ -321,7 +319,7 @@ \end{equation*} \end{thm} -Proposition \ref{thm:skein-modules} is immediate from the definition, and +Theorem \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 $\bc_*(S^1; -)$. @@ -384,16 +382,13 @@ \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}} +We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. \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}). +Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. Then \[ \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). @@ -436,8 +431,7 @@ \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}.} +Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. \begin{thm}[Higher dimensional Deligne conjecture] @@ -446,9 +440,7 @@ 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.} - - +\nn{Explain and sketch} %% == end of paper: diff -r 8378e03d3c7f -r e5ab1b074d88 pnas/preamble.tex --- a/pnas/preamble.tex Mon Oct 25 13:36:12 2010 -0700 +++ b/pnas/preamble.tex Mon Oct 25 14:01:33 2010 -0700 @@ -37,7 +37,7 @@ \newcommand{\tensor}{\otimes} \newcommand{\Tensor}{\bigotimes} -\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{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.39,-0.16); \draw[->] (0,0.2) .. controls (0.5,0.6) and (0.5,-0.4) .. (0,0);}}}} \newcommand{\bdy}{\partial} \newcommand{\compose}{\circ}