--- a/pnas/pnas.tex	Mon Nov 08 10:23:10 2010 +0900
+++ b/pnas/pnas.tex	Wed Nov 10 10:40:29 2010 +0900
@@ -226,11 +226,8 @@
 Note that the functoriality in the above axiom allows us to operate via
 homeomorphisms which are not the identity on the boundary of the $k$-ball.
 The action of these homeomorphisms gives the ``strong duality" structure.
-
-Next we consider domains and ranges of $k$-morphisms.
-Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism
-into domain and range --- the duality operations can convert domain to range and vice-versa.
-Instead, we will use a unified domain/range, which we will call a ``boundary".
+As such, we don't subdivide the boundary of a morphism
+into domain and range --- the duality operations can convert between domain and range.
 
 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
 
@@ -239,11 +236,11 @@
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
 
-For $c\in \cl{\cC}_{k-1}(\bd X)$ we let $\cC_k(X; c)$ denote the preimage $\bd^{-1}(c)$.
+For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$.
 
 Many of the examples we are interested in are enriched in some auxiliary category $\cS$
-(e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
-This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
+(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
+This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
 of an object of $\cS$, and all of the structure maps of the category (above and below) are
 compatible with the $\cS$ structure on $\cC_n(X; c)$.
 
@@ -298,6 +295,16 @@
 \label{axiom:product}
 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
 there is a map $\pi^*:\cC(X)\to \cC(E)$.
+These maps must be
+\begin{enumerate}
+\item natural with respect to maps of pinched products,
+\item functorial with respect to composition of pinched products, 
+\item compatible with gluing and restriction of pinched products.
+\end{enumerate}
+
+%%% begin noop %%%
+% this was the original list of conditions, which I've replaced with the much terser list above -S
+\noop{
 These maps must satisfy the following conditions.
 \begin{enumerate}
 \item
@@ -338,6 +345,7 @@
 	\res_D\circ\pi^* = \rho^*\circ\res_Y .
 \]
 \end{enumerate}
+} %%% end \noop %%%
 \end{axiom}
 \begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
@@ -676,7 +684,9 @@
 
 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
 
-\todo{Sketch proof}
+\begin{proof}
+We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and functoriality.
+\end{proof} 
 
 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
 \[
@@ -703,7 +713,7 @@
 %% \appendix[Appendix Title]
 
 \begin{acknowledgments}
--- text of acknowledgments here, including grant info --
+\nn{say something here}
 \end{acknowledgments}
 
 %% PNAS does not support submission of supporting .tex files such as BibTeX.
@@ -756,16 +766,44 @@
 
 
 \begin{figure}
+\centering
+\begin{tikzpicture}[%every label/.style={green}
+]
+\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
+\draw (S) arc  (-90:90:1);
+\draw (N) arc  (90:270:1);
+\node[left] at (-1,1) {$B_1$};
+\node[right] at (1,1) {$B_2$};
+\end{tikzpicture}
+\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure}
+
+\begin{figure}
+\centering
+\begin{tikzpicture}[%every label/.style={green},
+				x=1.5cm,y=1.5cm]
+\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
+\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
+\draw (S) arc  (-90:90:1);
+\draw (N) arc  (90:270:1);
+\draw (N) -- (S);
+\node[left] at (-1/4,1) {$B_1$};
+\node[right] at (1/4,1) {$B_2$};
+\node at (1/6,3/2)  {$Y$};
+\end{tikzpicture}
+\caption{From two balls to one ball.}\label{blah5}\end{figure}
+
+\begin{figure}
 \begin{equation*}
 \mathfig{.23}{ncat/zz2}
 \end{equation*}
-\caption{A small part of $\cell(W)$}
+\caption{A small part of $\cell(W)$.}
 \label{partofJfig}
 \end{figure}
 
 \begin{figure}
 $$\mathfig{.4}{deligne/manifolds}$$
-\caption{An $n$-dimensional surgery cylinder}\label{delfig2}
+\caption{An $n$-dimensional surgery cylinder.}\label{delfig2}
 \end{figure}