
diff -r 1b441b3595c9 -r 8599e156a169 blob1.tex
--- a/blob1.tex	Tue Mar 04 16:33:23 2008 +0000
+++ b/blob1.tex	Mon Apr 21 17:41:17 2008 +0000
@@ -4,7 +4,7 @@
 
 \usepackage[all]{xy}
 
-% test edit #2
+% test edit #3
 
 %%%%% excerpts from my include file of standard macros
 
@@ -163,14 +163,20 @@
 interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
 of the algebra.
 
-For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
+\medskip
+
+For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)
+that are common in the literature.
+We describe these carefully here.
+
+A field on a 0-manifold $P$ is a labeling of each point of $P$ with
 an object of the 2-category $C$.
 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
 A field on a 2-manifold $Y$ consists of
 \begin{itemize}
 	\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
 that each component of the complement is homeomorphic to a disk);
-	\item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$)
+	\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
 by a 0-morphism of $C$;
 	\item a transverse orientation of each 1-cell, thought of as a choice of
 ``domain" and ``range" for the two adjacent 2-cells;
@@ -195,9 +201,11 @@
 domain and range determined by the labelings of the link of $j$-cell.
 \end{itemize}
 
-\nn{next definition might need some work; I think linearity relations should
-be treated differently (segregated) from other local relations, but I'm not sure
-the next definition is the best way to do it}
+%\nn{next definition might need some work; I think linearity relations should
+%be treated differently (segregated) from other local relations, but I'm not sure
+%the next definition is the best way to do it}
+
+\medskip
 
 For top dimensional ($n$-dimensional) manifolds, we're actually interested 
 in the linearized space of fields.
@@ -245,7 +253,10 @@
 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ 
 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
 
-\nn{implies (extended?) isotopy; stable under gluing; open covers?; ...}
+\nn{Roughly, these are (1) the local relations imply (extended) isotopy; 
+(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
+(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). 
+See KW TQFT notes for details.  Need to transfer details to here.}
 
 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
@@ -259,6 +270,8 @@
 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$,
 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$.
 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$.
+\nn{Is this true in high (smooth) dimensions?  Self-diffeomorphisms of $B^n$
+rel boundary might not be isotopic to the identity.  OK for PL and TOP?}
 
 Given a system of fields and local relations, we define the skein space
 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on
@@ -316,7 +329,7 @@
 just erasing the blob from the picture
 (but keeping the blob label $u$).
 
-Note that the skein module $A(X)$
+Note that the skein space $A(X)$
 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
 
 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
@@ -401,7 +414,7 @@
 
 (Alert readers will have noticed that for $k=2$ our definition
 of $\bc_k(X)$ is slightly different from the previous definition
-of $\bc_2(X)$.
+of $\bc_2(X)$ --- we did not impose the reordering relations.
 The general definition takes precedence;
 the earlier definition was simplified for purposes of exposition.)
 
@@ -424,8 +437,9 @@
 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
 
 
-\nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just
-a vector space; relations to Chas-Sullivan string stuff}
+\nn{TO DO: 
+expand definition to handle DGA and $A_\infty$ versions of $n$-categories; 
+relations to Chas-Sullivan string stuff}
 
 
 
@@ -493,6 +507,21 @@
 \end{prop}
 
 
+% oops -- duplicate
+
+%\begin{prop} \label{functorialprop}
+%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of
+%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms.
+%\end{prop}
+
+%\begin{proof}
+%Obvious.
+%\end{proof}
+
+%\nn{need to same something about boundaries and boundary conditions above.
+%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
+
+
 \begin{prop}
 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
 of $n$-manifolds and diffeomorphisms to the category of chain complexes and 
@@ -500,6 +529,9 @@
 \qed
 \end{prop}
 
+\nn{need to same something about boundaries and boundary conditions above.
+maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
+
 
 In particular,
 \begin{prop}  \label{diff0prop}




