
diff -r 46b5c4f3e83c -r 4093d7979c56 blob1.tex
--- a/blob1.tex	Fri Feb 27 03:29:41 2009 +0000
+++ b/blob1.tex	Sat Feb 28 16:00:38 2009 +0000
@@ -378,6 +378,23 @@
 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
 \end{enumerate}
 
+\bigskip
+Using the functoriality and $\bullet\times I$ properties above, together
+with boundary collar homeomorphisms of manifolds, we can define the notion of 
+{\it extended isotopy}.
+Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
+of $\bd M$.
+Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$.
+Let $c$ be $x$ restricted to $Y$.
+Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
+Then we have the glued field $x \cup (c\times I)$ on $M \cup (Y\times I)$.
+Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
+Then we say that $x$ is {\it extended isotopic} to $f(x \cup (c\times I))$.
+More generally, we define extended isotopy to be the equivalence relation on fields
+on $M$ generated by isotopy plus all instance of the above construction
+(for all appropriate $Y$ and $x$).
+
+\nn{should also say something about pseudo-isotopy}
 
 \bigskip
 \hrule
@@ -521,33 +538,32 @@
 \subsection{Local relations}
 \label{sec:local-relations}
 
-\nn{the following is not done yet}
 
-Let $B^n$ denote the standard $n$-ball.
-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 two properties.
+A {\it local relation} is a collection subspaces $U(B; c) \sub \c[\cC_l(B; c)]$
+(for all $n$-manifolds $B$ which are
+homeomorphic to the standard $n$-ball and 
+all $c \in \cC(\bd B)$) satisfying the following properties.
 \begin{enumerate}
-\item local relations imply (extended) isotopy \nn{...}
-\item $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing \nn{...}
+\item functoriality: 
+$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
+\item local relations imply extended isotopy: 
+if $x, y \in \cC(B; c)$ and $x$ is extended isotopic 
+to $y$, then $x-y \in U(B; c)$.
+\item ideal with respect to gluing:
+if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\cup r \in U(B)$
 \end{enumerate}
 See \cite{kw:tqft} for details.
 
 
-For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
+For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \cC_l(B; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
 
-For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map
-$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
+$\cC_l(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
 domain and range.
 
 \nn{maybe examples of local relations before general def?}
 
-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
 the $n$-manifold $Y$ modulo local relations.




