Binary file diagrams/tempkw/jun23d.pdf has changed
--- a/sandbox.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/sandbox.tex Mon Jul 19 15:38:35 2010 -0600
@@ -11,26 +11,6 @@
\title{Sandbox}
\begin{document}
-$$
-\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
-\draw (0,0) node(R) {}
- -- (0.75,0) node[below] {$\bar{B}$}
- --(1.5,0) node[circle,fill=black,inner sep=2pt] {}
- arc (0:80:1.5) node[above] {$D \times I$}
- arc (80:180:1.5);
-\foreach \r in {0.3, 0.6, 0.9, 1.2} {
- \draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r);
-}
-\draw[fill=white]
- (R) node[circle,fill=black,inner sep=2pt] {}
- arc (45:65:3) node[below] {$B$}
- arc (65:90:3) node[below] {$A$}
- arc (90:135:3) node[circle,fill=black,inner sep=2pt] {}
- arc (-135:-90:3) node[below] {$C$}
- arc (-90:-45:3);
-\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {$D$};
-\node at (-2,0) {\scalebox{2.0}{$\uparrow f$}};
-\node at (0.2,0.8) {\scalebox{2.0}{$\uparrow \psi$}};
-\end{tikzpicture}
-$$
+
+
\end{document}
--- a/text/appendixes/smallblobs.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/appendixes/smallblobs.tex Mon Jul 19 15:38:35 2010 -0600
@@ -16,7 +16,7 @@
\end{rem}
\begin{proof}
This follows from Remark \ref{rem:for-small-blobs} following the proof of
-Proposition \ref{CHprop}.
+Theorem \ref{thm:CH}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:small-blobs}]
@@ -24,10 +24,10 @@
We will construct a chain map $s: \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=i\circ s - \id$. The composition $s \circ i$ will just be the identity.
On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover.
-\nn{KW: For some systems of fields this is not true.
-For example, consider a planar algebra with boxes of size greater than zero.
-So I think we should do the homotopy even in degree zero.
-But as noted above, maybe it's best to ignore this.}
+%\nn{KW: For some systems of fields this is not true.
+%For example, consider a planar algebra with boxes of size greater than zero.
+%So I think we should do the homotopy even in degree zero.
+%But as noted above, maybe it's best to ignore this.}
Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ ``makes $\beta$ small" if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism ``makes $\beta$ $\epsilon$-small" if the image of each ball is contained in some open ball of radius $\epsilon$.
@@ -119,7 +119,7 @@
It may be useful to look at Figure \ref{fig:erectly-a-tent-badly} to help understand the arrangement. The red, blue and orange $2$-cells there correspond to the $m=0$, $m=1$ and $m=2$ terms respectively, while the $3$-cells (only one of each type is shown) correspond to the terms in the homotopy $h$.
\begin{figure}[!ht]
$$\mathfig{0.5}{smallblobs/tent}$$
-\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms. \nn{turn upside?}}
+\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms.}
\label{fig:erectly-a-tent-badly}
\end{figure}
--- a/text/blobdef.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/blobdef.tex Mon Jul 19 15:38:35 2010 -0600
@@ -127,6 +127,14 @@
\nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below}
}
+\begin{defn}
+An \emph{$n$-ball decomposition} of a topological space $X$ is
+finite collection of triples $\{(B_i, X_i, Y_i)\}_{i=1,\ldots, k}$ where $B_i$ is an $n$-ball, $X_i$ is some topological space, and $Y_i$ is pair of disjoint homeomorphic $n-1$-manifolds in the boundary of $X_{i-1}$ (for convenience, $X_0 = Y_1 = \eset$), such that $X_{i+1} = X_i \cup_{Y_i} B_i$ and $X = X_k \cup_{Y_k} B_k$.
+
+Equivalently, we can define a ball decomposition inductively. A ball decomposition of $X$ is a topological space $X'$ along with a pair of disjoint homeomorphic $n-1$-manifolds $Y \subset \bdy X$, so $X = X' \bigcup_Y \selfarrow$, and $X'$ is either a disjoint union of balls, or a topological space equipped with a ball decomposition.
+\end{defn}
+
+
Before describing the general case we should say more precisely what we mean by
disjoint and nested blobs.
Disjoint will mean disjoint interiors.
--- a/text/evmap.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/evmap.tex Mon Jul 19 15:38:35 2010 -0600
@@ -36,9 +36,13 @@
}
\end{equation*}
\end{enumerate}
-Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
-satisfying the above two conditions.
+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.
\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.
+\end{rem}
+
Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma},
and then give an outline of the method of proof.
@@ -345,7 +349,7 @@
\begin{proof}
-There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ 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
@@ -582,9 +586,6 @@
these two maps agree up to $m$-th order homotopy.
More precisely, one can show that the subcomplex of maps containing the various
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$.
-\nn{now should remark that we have not, in fact, produced a contractible set of maps,
-but we have come very close}
-\nn{better: change statement of thm}
\medskip
--- a/text/intro.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/intro.tex Mon Jul 19 15:38:35 2010 -0600
@@ -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} \\
--- a/text/ncat.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/ncat.tex Mon Jul 19 15:38:35 2010 -0600
@@ -821,8 +821,7 @@
This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
Notice that with $F$ a point, the above example is a construction turning a topological
-$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
-\nn{do we use this notation elsewhere (anymore)?}
+$n$-category $\cC$ into an $A_\infty$ $n$-category.
We think of this as providing a ``free resolution"
of the topological $n$-category.
\nn{say something about cofibrant replacements?}
@@ -1025,8 +1024,6 @@
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit
is more involved.
-\nn{should change to less strange terminology: ``filtration" to ``simplex"
-(search for all occurrences of ``filtration")}
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.
Such sequences (for all $m$) form a simplicial set in $\cell(W)$.
Define $\cl{\cC}(W)$ as a vector space via
@@ -1034,9 +1031,6 @@
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,
\]
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$.
-(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$,
-the complex $U[m]$ is concentrated in degree $m$.)
-\nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?}
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$
summands plus another term using the differential of the simplicial set of $m$-sequences.
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
@@ -1051,14 +1045,13 @@
%combine only two balls at a time; for $n=1$ this version will lead to usual definition
%of $A_\infty$ category}
-We will call $m$ the filtration degree of the complex.
-\nn{is there a more standard term for this?}
+We will call $m$ the simplex degree of the complex.
We can think of this construction as starting with a disjoint copy of a complex for each
-permissible decomposition (filtration degree 0).
+permissible decomposition (simplex degree 0).
Then we glue these together with mapping cylinders coming from gluing maps
-(filtration degree 1).
+(simplex degree 1).
Then we kill the extra homology we just introduced with mapping
-cylinders between the mapping cylinders (filtration degree 2), and so on.
+cylinders between the mapping cylinders (simplex degree 2), and so on.
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.
@@ -2220,9 +2213,80 @@
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$.
(See Figure \ref{jun23d}.)
\begin{figure}[t]
-\begin{equation*}
-\mathfig{.9}{tempkw/jun23d}
-\end{equation*}
+\begin{tikzpicture}
+\node(L) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw[red] (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw (1.5,0) arc (0:149:1.5);
+\draw[red]
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw (R) arc (45:75:3);
+\draw (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\node(M) at (5,4) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw[red] (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw(1.5,0) arc (0:149:1.5);
+\draw
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw[red] (R) arc (45:75:3);
+\draw[red] (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\node(R) at (10,0) {
+\scalebox{0.5}{
+\begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm]
+\draw[red] (0.75,0) -- +(2,0);
+\draw (0,0) node(R) {}
+ -- (0.75,0) node[below] {}
+ --(1.5,0) node[circle,fill=black,inner sep=2pt] {};
+\draw[fill] (150:1.5) circle (2pt) node[above=4pt] {};
+\draw[red] (1.5,0) arc (0:149:1.5);
+\draw
+ (R) node[circle,fill=black,inner sep=2pt] {}
+ arc (-45:-135:3) node[circle,fill=black,inner sep=2pt] {};
+\draw[red] (-5.5,0) -- (-4.2,0);
+\draw (R) arc (45:75:3);
+\draw[red] (150:1.5) arc (74:135:3);
+\node at (-2,0) {\scalebox{2.0}{$B_1$}};
+\node at (0.2,0.8) {\scalebox{2.0}{$B_2$}};
+\node at (-4,1.2) {\scalebox{2.0}{$A$}};
+\node at (-4,-1.2) {\scalebox{2.0}{$C$}};
+\node[red] at (2.53,0.35) {\scalebox{2.0}{$E$}};
+\end{tikzpicture}
+}
+};
+\draw[->] (L) to[out=90,in=225] node[sloped, above] {push $B_1$} (M);
+\draw[->] (M) to[out=-45,in=90] node[sloped, above] {push $B_2$} (R);
+\draw[->] (L) to[out=-35,in=-145] node[sloped, below] {push $B_1 \cup B_2$} (R);
+\end{tikzpicture}
\caption{A movie move}
\label{jun23d}
\end{figure}
--- a/text/tqftreview.tex Mon Jul 19 15:38:18 2010 -0600
+++ b/text/tqftreview.tex Mon Jul 19 15:38:35 2010 -0600
@@ -111,9 +111,9 @@
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.
+$W$ might intersect along their boundaries. \todo{Really? I thought we wanted the boundaries of the two copies of Y to be disjoint}
Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$
-(Figure xxxx).
+(Figure \ref{fig:???}).
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself
(without corners) along two copies of $\bd Y$.
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let
@@ -245,9 +245,7 @@
One of the advantages of string diagrams over pasting diagrams is that one has more
flexibility in slicing them up in various ways.
In addition, string diagrams are traditional in quantum topology.
-The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{
-MR0281657,MR776784 % penrose
-}
+The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{MR0281657,MR776784} % both penrose
If $X$ has boundary, we require that the cell decompositions are in general
position with respect to the boundary --- the boundary intersects each cell