--- a/text/blobdef.tex Thu Jul 22 09:48:51 2010 -0700
+++ b/text/blobdef.tex Thu Jul 22 12:20:42 2010 -0600
@@ -5,7 +5,8 @@
Let $X$ be an $n$-manifold.
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
+We'll assume it is enriched over \textbf{Vect};
+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. by writing $\lf(X)$ instead of $\lf(X; c)$.
@@ -20,14 +21,24 @@
\]
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
-In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way, then explain the technical difficulties, and finally give a cumbersome but complete definition in Definition \ref{defn:blobs}. If (we don't recommend it) you want to keep track of the ways in which this initial description is misleading, or you're reading through a second time to understand the technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'', ``nested'' and ``disjoint'', that are not quite the intuitive ones. Moreover some of the pieces into which we cut manifolds below are not themselves manifolds, and it requires special attention to define fields on these pieces.
+In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way,
+then explain the technical difficulties, and finally give a cumbersome but complete definition in
+Definition \ref{defn:blobs}.
+If (we don't recommend it) you want to keep track of the ways in which
+this initial description is misleading, or you're reading through a second time to understand the
+technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'',
+``nested'' and ``disjoint'', that are not quite the intuitive ones.
+Moreover some of the pieces
+into which we cut manifolds below are not themselves manifolds, and it requires special attention
+to define fields on these pieces.
We of course define $\bc_0(X) = \lf(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf{\bdy X}$.
-We'll omit this sort of detail in the rest of this section.)
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf(\bdy X)$.
+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 $\bc_0(X)$'.
+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:
\begin{itemize}
\item An closed ball in $X$ (``blob") $B \sub X$.
@@ -56,6 +67,7 @@
just erasing the blob from the picture
(but keeping the blob label $u$).
+\nn{it seems rather strange to make this a theorem}
Note that directly from the definition we have
\begin{thm}
\label{thm:skein-modules}
@@ -64,9 +76,9 @@
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
+Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations
(redundancies, syzygies) among the
-local relations encoded in $\bc_1(X)$'.
+local relations encoded in $\bc_1(X)$''.
A $2$-blob diagram, comes in one of two types, disjoint and nested.
A disjoint 2-blob diagram consists of
\begin{itemize}
@@ -116,11 +128,15 @@
\right) \bigoplus \\
&& \quad\quad \left(
\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2}
- U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2)
+ U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1, c_2) \tensor \cC(X \setminus B_2; c_2)
\right) .
\end{eqnarray*}
-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).
+% __ (already said this above)
+%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).
+
+
+
Before describing the general case, note that when we say blobs are disjoint, we will only mean that their interiors are disjoint. Nested blobs may have boundaries that overlap, or indeed may coincide.
A $k$-blob diagram consists of
--- a/text/comm_alg.tex Thu Jul 22 09:48:51 2010 -0700
+++ b/text/comm_alg.tex Thu Jul 22 12:20:42 2010 -0600
@@ -3,14 +3,10 @@
\section{Commutative algebras as $n$-categories}
\label{sec:comm_alg}
-\nn{should consider leaving this out; for now, make it an appendix.}
-
-\nn{also, this section needs a little updating to be compatible with the rest of the paper.}
-
If $C$ is a commutative algebra it
can also be thought of as an $n$-category whose $j$-morphisms are trivial for
$j<n$ and whose $n$-morphisms are $C$.
-The goal of this \nn{subsection?} is to compute
+The goal of this appendix is to compute
$\bc_*(M^n, C)$ for various commutative algebras $C$.
Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative
@@ -35,24 +31,24 @@
\end{prop}
\begin{proof}
-To define the chain maps between the two complexes we will use the following lemma:
-
-\begin{lemma}
-Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
-a basis (e.g.\ blob diagrams or singular simplices).
-For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
-such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
-Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
-$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
-\end{lemma}
-
-\begin{proof}
-\nn{easy, but should probably write the details eventually}
-\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
-\end{proof}
-
+%To define the chain maps between the two complexes we will use the following lemma:
+%
+%\begin{lemma}
+%Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
+%a basis (e.g.\ blob diagrams or singular simplices).
+%For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
+%such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
+%Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
+%$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
+%\end{lemma}
+%
+%\begin{proof}
+%\nn{easy, but should probably write the details eventually}
+%\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
+%\end{proof}
+We will use acyclic models \nn{need ref}.
Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
-satisfying the conditions of the above lemma.
+satisfying the conditions of \nn{need ref}.
If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a
finite unordered collection of points of $M$ with multiplicities, which is
a point in $\Sigma^\infty(M)$.
@@ -66,7 +62,8 @@
and using this point we can embed $X$ in $\Sigma^\infty(M)$.
Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a
subspace of $\Sigma^\infty(M)$.
-It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma.
+It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from
+\nn{need ref, or state condition}.
Thus we have defined (up to homotopy) a map from
$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$.
@@ -84,7 +81,7 @@
Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter.
\nn{need to say more precisely how small}
Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$.
-This is contractible by \ref{bcontract}.
+This is contractible by Proposition \ref{bcontract}.
We can arrange that the boundary/inclusion condition is satisfied if we start with
low-dimensional simplices and work our way up.
\nn{need to be more precise}
--- a/text/tqftreview.tex Thu Jul 22 09:48:51 2010 -0700
+++ b/text/tqftreview.tex Thu Jul 22 12:20:42 2010 -0600
@@ -110,8 +110,8 @@
We say that fields on $X\sgl$ in the image of the gluing map
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. \todo{Really? I thought we wanted the boundaries of the two copies of Y to be disjoint}
+Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$
+are disjoint from each other and $\bd(Y\du Y) = \bd W$.
Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
(Figure \ref{fig:???}).
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself