
diff -r 57291331fd82 -r e0b304e6b975 text/definitions.tex
--- a/text/definitions.tex	Tue Oct 27 22:27:07 2009 +0000
+++ b/text/definitions.tex	Wed Oct 28 00:54:35 2009 +0000
@@ -6,6 +6,11 @@
 In this section we review the construction of TQFTs from ``topological fields".
 For more details see xxxx.
 
+We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
+submanifold of $X$, then $X \setmin Y$ implicitly means the closure
+$\overline{X \setmin Y}$.
+
+
 \subsection{Systems of fields}
 \label{sec:fields}
 
@@ -346,13 +351,18 @@
 In this section we will usually suppress boundary conditions on $X$ from the notation
 (e.g. write $\lf(X)$ instead of $\lf(X; c)$).
 
-We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
-submanifold of $X$, then $X \setmin Y$ implicitly means the closure
-$\overline{X \setmin Y}$.
+We want to replace the quotient
+\[
+	A(X) \deq \lf(X) / U(X)
+\]
+of the previous section with a resolution
+\[
+	\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .
+\]
 
 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
 
-Define $\bc_0(X) = \lf(X)$.
+We of course define $\bc_0(X) = \lf(X)$.
 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
 We'll omit this sort of detail in the rest of this section.)
 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
@@ -367,6 +377,10 @@
 \item A local relation field $u \in U(B; c)$
 (same $c$ as previous bullet).
 \end{itemize}
+(See Figure \ref{blob1diagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob1diagram}
+\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
 In order to get the linear structure correct, we (officially) define
 \[
 	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
@@ -400,6 +414,10 @@
 (where $c_i \in \cC(\bd B_i)$).
 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.
 \end{itemize}
+(See Figure \ref{blob2ddiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob2ddiagram}
+\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
 reversing the order of the blobs changes the sign.
 Define $\bd(B_0, B_1, u_0, u_1, r) = 
@@ -416,6 +434,10 @@
 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
 \item A local relation field $u_0 \in U(B_0; c_0)$.
 \end{itemize}
+(See Figure \ref{blob2ndiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob2ndiagram}
+\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
 (for some $c_1 \in \cC(B_1)$) and
 $r' \in \cC(X \setmin B_1; c_1)$.
@@ -427,8 +449,6 @@
 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
 It is again easy to check that $\bd^2 = 0$.
 
-\nn{should draw figures for 1, 2 and $k$-blob diagrams}
-
 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
 (officially)
 \begin{eqnarray*}
@@ -469,6 +489,10 @@
 where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
 If $B_i = B_j$ then $u_i = u_j$.
 \end{itemize}
+(See Figure \ref{blobkdiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blobkdiagram}
+\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
 
 If two blob diagrams $D_1$ and $D_2$ 
 differ only by a reordering of the blobs, then we identify




