--- 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.