# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1279554182 25200
# Node ID 3377d4db80d9b4ca94db42f673de39e5aa081a56
# Parent  e88e44347b36626b1c882ec264439ccd5e791ba2
removing superscripts from \bc

diff -r e88e44347b36 -r 3377d4db80d9 text/intro.tex
--- a/text/intro.tex	Mon Jul 19 08:42:24 2010 -0700
+++ b/text/intro.tex	Mon Jul 19 08:43:02 2010 -0700
@@ -209,12 +209,12 @@
 That is, 
 for a fixed $n$-dimensional system of fields $\cC$, the association
 \begin{equation*}
-X \mapsto \bc_*^{\cC}(X)
+X \mapsto \bc_*(X; \cC)
 \end{equation*}
 is a functor from $n$-manifolds and homeomorphisms between them to chain 
 complexes and isomorphisms between them.
 \end{property}
-As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; 
+As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; 
 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.
 
 The blob complex is also functorial (indeed, exact) with respect to $\cC$, 
@@ -250,7 +250,7 @@
 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
 \begin{equation*}
-\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
+\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation*}
 \end{property}
 
@@ -271,7 +271,7 @@
 by $\cC$.
 (See \S \ref{sec:local-relations}.)
 \begin{equation*}
-H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
+H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm:skein-modules}
 
@@ -281,7 +281,7 @@
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{equation*}
-\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
+\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
 \end{thm:hochschild}
 
@@ -297,8 +297,7 @@
 
 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}}
 
-\begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\
-\vspace{-0.5cm}
+\begin{thm:CH}[$C_*(\Homeo(-))$ action]
 \label{thm:evaluation}%
 There is a chain map
 \begin{equation*}
@@ -313,10 +312,10 @@
 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).
 \begin{equation*}
 \xymatrix@C+2cm{
-     \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow) \\
      \CH{X} \otimes \bc_*(X)
-        \ar[r]_{\ev_{X}}  \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
-            \bc_*(X) \ar[u]_{\gl_Y}
+        \ar[r]_{\ev_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
+            \bc_*(X) \ar[d]_{\gl_Y} \\
+     \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow)
 }
 \end{equation*}
 \end{enumerate}
@@ -329,7 +328,7 @@
 
 Further,
 \begin{thm:CH-associativity}
-\item The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy).
+The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy).
 \begin{equation*}
 \xymatrix{
 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\
diff -r e88e44347b36 -r 3377d4db80d9 text/tqftreview.tex
--- a/text/tqftreview.tex	Mon Jul 19 08:42:24 2010 -0700
+++ b/text/tqftreview.tex	Mon Jul 19 08:43:02 2010 -0700
@@ -111,9 +111,9 @@
 are transverse to $Y$ or splittable along $Y$.
 \item Gluing with corners.
 Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and 
-$W$ might intersect along their boundaries.
+$W$ might intersect along their boundaries. \todo{Really? I thought we wanted the boundaries of the two copies of Y to be disjoint}
 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
-(Figure xxxx).
+(Figure \ref{fig:???}).
 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
 (without corners) along two copies of $\bd Y$.
 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let
@@ -245,9 +245,7 @@
 One of the advantages of string diagrams over pasting diagrams is that one has more
 flexibility in slicing them up in various ways.
 In addition, string diagrams are traditional in quantum topology.
-The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{
-MR0281657,MR776784 % penrose
-}
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{MR0281657,MR776784} % both penrose
 
 If $X$ has boundary, we require that the cell decompositions are in general
 position with respect to the boundary --- the boundary intersects each cell