--- a/text/blobdef.tex Sun Sep 19 23:15:21 2010 -0700
+++ b/text/blobdef.tex Mon Sep 20 06:10:49 2010 -0700
@@ -67,7 +67,8 @@
just erasing the blob from the picture
(but keeping the blob label $u$).
-\nn{it seems rather strange to make this a theorem} \nn{it's a theorem because it's stated in the introduction, and I wanted everything there to have numbers that pointed into the paper --S}
+\nn{it seems rather strange to make this a theorem}
+\nn{it's a theorem because it's stated in the introduction, and I wanted everything there to have numbers that pointed into the paper --S}
Note that directly from the definition we have
\begin{thm}
\label{thm:skein-modules}
@@ -137,7 +138,10 @@
\medskip
-Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested, along with fields on all the components that the blobs divide $X$ into. Blobs which have no other blobs inside are called `twig blobs', and the fields on the twig blobs must be local relations.
+Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested,
+along with fields on all the components that the blobs divide $X$ into.
+Blobs which have no other blobs inside are called `twig blobs',
+and the fields on the twig blobs must be local relations.
The boundary is the alternating sum of erasing one of the blobs.
In order to describe this general case in full detail, we must give a more precise description of
which configurations of balls inside $X$ we permit.
@@ -162,7 +166,11 @@
C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\
D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}.
\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 $[-1,1]^2 \times [0,1]$. Note however that the complement is not a manifold.
+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 $[-1,1]^2 \times [0,1]$.
+Note however that the complement is not a manifold.
\end{example}
\begin{defn}
@@ -172,7 +180,8 @@
by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$.
If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}.
\end{defn}
-Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$.
+Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is
+splittable along it if it is the image of a field on $M_0$.
In the example above, note that
\[
@@ -203,11 +212,18 @@
We describe these as disjoint blobs and nested blobs.
Note that nested blobs may have boundaries that overlap, or indeed coincide.
Blobs may meet the boundary of $X$.
-Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together.
+Further, note that blobs need not actually be embedded balls in $X$, since parts of the
+boundary of the ball $M_r'$ may have been glued together.
-Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, we can just take $M_0$ to be these pieces, and $M_1 = X$.
+Note that often the gluing decomposition for a configuration of blobs may just be the trivial one:
+if the boundaries of all the blobs cut $X$ into pieces which are all manifolds,
+we can just take $M_0$ to be these pieces, and $M_1 = X$.
-In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are
+In the informal description above, in the definition of a $k$-blob diagram we asked for any
+collection of $k$ balls which were pairwise disjoint or nested.
+We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}.
+Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs;
+this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are
\begin{defn}
\label{defn:blob-diagram}
A $k$-blob diagram on $X$ consists of
@@ -216,7 +232,11 @@
\item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration,
\end{itemize}
such that
-the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cF(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$.
+the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace
+$U(B_i) \subset \cF(B_i)$.
+(See Figure \ref{blobkdiagram}.)
+More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above,
+and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$.
\end{defn}
\begin{figure}[t]\begin{equation*}
\mathfig{.7}{definition/k-blobs}
@@ -224,13 +244,22 @@
and
\begin{defn}
\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 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 $r$:
+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
+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 $r$:
\begin{equation*}
\bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r)
\end{equation*}
\end{defn}
-We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams.
-A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components.
+We readily see that if a gluing decomposition is compatible with some configuration of blobs,
+then it is also compatible with any configuration obtained by forgetting some blobs,
+ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams.
+A slight compensation to the complication of the official definition arising from attention
+to splitting is that the differential now just preserves the entire field $r$ without
+having to say anything about gluing together fields on smaller components.
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms,
is immediately obvious from the definition.
@@ -257,11 +286,14 @@
\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.
-(When the fields come from an $n$-category, this correspondence works best if we think of each twig label $u_i$ as having the form
+(When the fields come from an $n$-category, 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 $B_i$, $e: \cF(B_i) \to C$ is the evaluation map,
and $s:C \to \cF(B_i)$ is some fixed section of $e$.)
-For lack of a better name, we'll call elements of $P$ cone-product polyhedra,
+For lack of a better name,
+\nn{can we think of a better name?}
+we'll call elements of $P$ cone-product polyhedra,
and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set).
\end{remark}
--- a/text/evmap.tex Sun Sep 19 23:15:21 2010 -0700
+++ b/text/evmap.tex Mon Sep 20 06:10:49 2010 -0700
@@ -80,7 +80,8 @@
\end{lemma}
\begin{proof}
-It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that
+It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated
+pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that
\[
(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))
\]
@@ -156,7 +157,8 @@
disjoint union of balls.
Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and
-also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
+also satisfying conditions specified below.
+\nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
As before, choose a sequence of collar maps $f_j$
such that each has support
contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms
@@ -223,7 +225,8 @@
\item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous.
\item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous,
where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on
-$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
+$\bc_0(B)$ comes from the generating set $\BD_0(B)$.
+\nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
\end{itemize}
We can summarize the above by saying that in the typical continuous family
@@ -270,7 +273,8 @@
Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$.
A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron.
-We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.}
+We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$.
+\nn{I found it pretty confusing to reuse the letter $r$ here.}
Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking
the same value (namely $r(y(p))$, for any $p\in P$).
Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams
@@ -302,7 +306,10 @@
&= x - r(x) + r(x) \\
&= x.
\end{align*}
-Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
+Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram,
+as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$
+\nn{explain why this is true?}
+and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
For $x\in \btc_{00}$ we have
\begin{align*}
@@ -517,10 +524,12 @@
\end{equation*}
\end{enumerate}
Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$
-satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy.
+satisfying the above two conditions which is $m$-connected. In particular,
+this means that the choice of chain map above is unique up to homotopy.
\end{thm}
\begin{rem}
-Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this.
+Note that the statement doesn't quite give uniqueness up to iterated homotopy.
+We fully expect that this should actually be the case, but haven't been able to prove this.
\end{rem}
@@ -712,8 +721,8 @@
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
%We also have that $\deg(b'') = 0 = \deg(p'')$.
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
-This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields
-differ by a local relation.
+This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility}
+and the fact that isotopic fields differ by a local relation.
Finally, define
\[
e(p\ot b) \deq x' \bullet p''(b'') .
@@ -829,7 +838,8 @@
\begin{proof}
-There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
+There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set
+$\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
(Here we are using the fact that the blobs are
piecewise smooth or piecewise-linear and that $\bd c$ is collared.)
We need to consider all such $c$ because all generators appearing in