--- 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} \\