--- a/pnas/pnas.tex Wed Nov 24 09:51:28 2010 -0700
+++ b/pnas/pnas.tex Wed Nov 24 09:57:41 2010 -0700
@@ -633,6 +633,9 @@
it evaluates to a zero $n$-morphism of $C$.
The next few paragraphs describe this in more detail.
+We will call a string diagram on a manifold a ``field".
+(See \cite{1009.5025} for a more general notion of field.)
+
We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}
if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that
each $B_i$ appears as a connected component of one of the $M_j$.
@@ -651,10 +654,10 @@
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
+ $s$ is the product of linear combinations of fields $s_i$ 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).
+ \item the $s_i$'s corresponding to the other pieces are single fields (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}
@@ -662,10 +665,8 @@
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 linear combinations of string diagrams on $W$.
+For $k=0$, the $0$-blob group is simply linear combinations of fields (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.