
diff -r 57bd9fab3827 -r 001fc6183d19 pnas/pnas.tex
--- a/pnas/pnas.tex	Mon Nov 22 17:55:32 2010 -0700
+++ b/pnas/pnas.tex	Mon Nov 22 19:42:06 2010 -0700
@@ -639,21 +639,34 @@
 each $B_i$ appears as a connected component of one of the $M_j$. 
 Note that this forces the balls to be pairwise either disjoint or nested. 
 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. 
-These pieces need not be manifolds, but they do automatically have permissible decompositions.
+These pieces need not be manifolds, 
+but they can be further subdivided into pieces which are manifolds
+and which fit into a permissible decomposition of $W$.
 
 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. 
 A $k$-blob diagram consists of
 \begin{itemize}
-\item a permissible collection of $k$ embedded balls, and
-\item for each resulting piece of $W$, a field,
+	\item a permissible collection of $k$ embedded balls, and
+	\item a linear combination $s$ of string diagrams on $W$,
 \end{itemize}
-such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. 
-We call such a field a ``null field on $B$".
+such that
+\begin{itemize}
+	\item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that
+	$s$ is the product of linear combinations of string diagrams $s_i$ on the initial pieces $X_i$ of the decomposition
+	(for fixed restrictions to the boundaries of the pieces),
+	\item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and
+	\item the $s_i$'s corresponding to the other pieces are single string diagrams (linear combinations with only one term).
+\end{itemize}
+%that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. 
+\nn{yech}
+We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$".
 
 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
 
+\nn{KW: I have not finished changng terminology from ``field" to ``string diagram"}
+
 We now spell this out for some small values of $k$. 
-For $k=0$, the $0$-blob group is simply fields on $W$. 
+For $k=0$, the $0$-blob group is simply linear combinations of string diagrams on $W$. 
 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. 
 The differential simply forgets the ball. 
 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball.




