--- a/text/basic_properties.tex Sun Feb 06 18:31:17 2011 -0800
+++ b/text/basic_properties.tex Sun Feb 06 20:54:10 2011 -0800
@@ -31,16 +31,16 @@
conditions to the notation.
Suppose that for all $c \in \cC(\bd B^n)$
-we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
+we have a splitting $s: H_0(\bc_*(B^n; c)) \to \bc_0(B^n; c)$
of the quotient map
-$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
+$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n; c))$.
For example, this is always the case if the coefficient ring is a field.
Then
\begin{prop} \label{bcontract}
-For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
+For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n; c) \to H_0(\bc_*(B^n; c))$
is a chain homotopy equivalence
-with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.
-Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.
+with inverse $s: H_0(\bc_*(B^n; c)) \to \bc_*(B^n; c)$.
+Here we think of $H_0(\bc_*(B^n; c))$ as a 1-step complex concentrated in degree 0.
\end{prop}
\begin{proof}
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map
@@ -67,8 +67,13 @@
This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}.
\end{proof}
-Recall the definition of the support of a blob diagram as the union of all the
-blobs of the diagram.
+%Recall the definition of the support of a blob diagram as the union of all the
+%blobs of the diagram.
+We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$,
+to be the union of the blobs of $b$.
+For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
+we define $\supp(y) \deq \bigcup_i \supp(b_i)$.
+
For future use we prove the following lemma.
\begin{lemma} \label{support-shrink}
--- a/text/blobdef.tex Sun Feb 06 18:31:17 2011 -0800
+++ b/text/blobdef.tex Sun Feb 06 20:54:10 2011 -0800
@@ -33,9 +33,11 @@
to define fields on these pieces.
We of course define $\bc_0(X) = \cF(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$.
+In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
+
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for $c \in \cF(\bdy X)$.
+The blob complex $\bc_*(X; c)$ will depend on a fixed boundary condition $c\in \cF(\bdy X)$.
We'll omit such boundary conditions from the notation in the rest of this section.)
-In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
We want the vector space $\bc_1(X)$ to capture
``the space of all local relations that can be imposed on $\bc_0(X)$".
@@ -148,8 +150,8 @@
\item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add
$D$ itself as an outermost blob.
(This is used in the proof of Proposition \ref{bcontract}.)
-\item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs
-on $X$ gives rise to a permissible configuration on $X'$.
+\item If $X\sgl$ is obtained from $X$ by gluing, then any permissible configuration of blobs
+on $X$ gives rise to a permissible configuration on $X\sgl$.
(This is necessary for Proposition \ref{blob-gluing}.)
\end{itemize}
Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not
@@ -166,8 +168,8 @@
\end{align*}
Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$.
Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$,
-and $\{C\}$ is a valid configuration of blobs in $C \cup D$,
-so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$.
+and $\{D\}$ is a valid configuration of blobs in $C \cup D$,
+so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$.
Note however that the complement is not a manifold.
\end{example}
@@ -244,7 +246,7 @@
\label{defn:blobs}
The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all
configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration,
-modulo identifying the vector spaces for configurations that only differ by a permutation of the balls
+modulo identifying the vector spaces for configurations that only differ by a permutation of the blobs
by the sign of that permutation.
The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of
forgetting one blob from the configuration, preserving the field $r$:
@@ -263,11 +265,6 @@
is immediately obvious from the definition.
A homeomorphism acts in an obvious way on blobs and on fields.
-We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$,
-to be the union of the blobs of $b$.
-For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
-we define $\supp(y) \deq \bigcup_i \supp(b_i)$.
-
\begin{remark} \label{blobsset-remark} \rm
We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,
but with simplices replaced by a more general class of combinatorial shapes.
--- a/text/tqftreview.tex Sun Feb 06 18:31:17 2011 -0800
+++ b/text/tqftreview.tex Sun Feb 06 20:54:10 2011 -0800
@@ -456,10 +456,10 @@
%$\bc_0(X) = \lf(X)$.
\begin{defn}
\label{defn:TQFT-invariant}
-The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is
+The TQFT invariant of $X$ associated to a system of fields $\cC$ and local relations $U$ is
$$A(X) \deq \lf(X) / U(X),$$
-where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$:
-$\cU(X)$ is generated by fields of the form $u\bullet r$, where
+where $U(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$:
+$U(X)$ is generated by fields of the form $u\bullet r$, where
$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
\end{defn}
The blob complex, defined in the next section,