--- a/text/blobdef.tex Mon Jul 12 17:29:25 2010 -0600
+++ b/text/blobdef.tex Wed Jul 14 11:06:11 2010 -0600
@@ -4,38 +4,37 @@
\label{sec:blob-definition}
Let $X$ be an $n$-manifold.
-Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations.
-(If $\cC$ is not enriched over Vect, we can make it so by allowing finite
-linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.)
+Let $\cC$ be a fixed system of fields and local relations.
+We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite
+linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.
-In this section we will usually suppress boundary conditions on $X$ from the notation
-(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
+In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$.
We want to replace the quotient
\[
A(X) \deq \lf(X) / U(X)
\]
-of the previous section with a resolution
+of Definition \ref{defn:TQFT-invariant} 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 $\bc_k(X)$.
+We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. \todo{create a numbered definition for the general case}
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 vector space of fields on $X$.
+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 $\bc_0(X)$'.
-Thus we say a $1$-blob diagram consists of
+Thus we say a $1$-blob diagram consists of:
\begin{itemize}
\item An embedded closed ball (``blob") $B \sub X$.
\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$.
\item A field $r \in \cC(X \setmin B; c)$.
\item A local relation field $u \in U(B; c)$.
\end{itemize}
-(See Figure \ref{blob1diagram}.)
+(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation.
\begin{figure}[t]\begin{equation*}
\mathfig{.6}{definition/single-blob}
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
@@ -56,15 +55,18 @@
just erasing the blob from the picture
(but keeping the blob label $u$).
-Note that the skein space $A(X)$
-is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-This is Theorem \ref{thm:skein-modules}, and also used in the second
+Note that directly from the definition we have
+\begin{thm}
+\label{thm:skein-modules}
+The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
+\end{thm}
+This also establishes the second
half of Property \ref{property:contractibility}.
Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations
(redundancies, syzygies) among the
local relations encoded in $\bc_1(X)$'.
-More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested.
+A $2$-blob diagram, comes in one of two types, disjoint and nested.
A disjoint 2-blob diagram consists of
\begin{itemize}
\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.
@@ -98,12 +100,11 @@
\mathfig{.6}{definition/nested-blobs}
\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$.
-Note that the requirement that
-local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$.
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
sum of the two ways of erasing one of the blobs.
When we erase the inner blob, the outer blob inherits the label $u\bullet r'$.
-It is again easy to check that $\bd^2 = 0$.
+It is again easy to check that $\bd^2 = 0$. Note that the requirement that
+local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$.
As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is
\begin{eqnarray*}
@@ -117,8 +118,8 @@
U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2)
\right) .
\end{eqnarray*}
-For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
-(rather than a new, linearly independent 2-blob diagram).
+For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign
+(rather than a new, linearly independent, 2-blob diagram).
\noop{
\nn{Hmm, I think we should be doing this for nested blobs too --
we shouldn't force the linear indexing of the blobs to have anything to do with
@@ -157,7 +158,7 @@
\item A field $r \in \cC(X \setmin B^t; c^t)$,
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
is determined by the $c_i$'s.
-$r$ is required to be splittable along the boundaries of all blobs, twigs or not.
+$r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.)
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
If $B_i = B_j$ then $u_i = u_j$.
@@ -171,12 +172,12 @@
differ only by a reordering of the blobs, then we identify
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
-$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
+Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams.
As before, the official definition is in terms of direct sums
of tensor products:
\[
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
- \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
+ \left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
\]
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs.
@@ -190,9 +191,9 @@
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
Let $E_j(b)$ denote the result of erasing the $j$-th blob.
If $B_j$ is not a twig blob, this involves only decrementing
-the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
+the indices of blobs $B_{j+1},\ldots,B_{k}$.
If $B_j$ is a twig blob, we have to assign new local relation labels
-if removing $B_j$ creates new twig blobs.
+if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created}
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
Finally, define
@@ -203,7 +204,7 @@
Thus we have a chain complex.
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition.
-A homeomorphism acts in an obvious on blobs and on fields.
+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 $b$.
@@ -225,8 +226,8 @@
\end{itemize}
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
a diagram of $k$ disjoint blobs corresponds to a $k$-cube.
-(This correspondence works best if we thing of each twig label $u_i$ as having the form
+(This correspondence works best if we think of each twig label $u_i$ as having the form
$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map,
-and $s:C \to \cC(B_i)$ is some fixed section of $e$.)
+and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})