# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1300201272 25200
# Node ID 3ae1a110873b43577e9db144c32cbe1f1ba9c4a6
# Parent  27cfae8f433033489aea7d8c56bd18e0e60766c6
add definition of collaring homeo, etc.

diff -r 27cfae8f4330 -r 3ae1a110873b text/ncat.tex
--- a/text/ncat.tex	Tue Mar 15 07:25:13 2011 -0700
+++ b/text/ncat.tex	Tue Mar 15 08:01:12 2011 -0700
@@ -537,8 +537,9 @@
 This axiom needs to be strengthened to force product morphisms to act as the identity.
 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
 Let $J$ be a 1-ball (interval).
-We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
-(Here we use $Y\times J$ with boundary entirely pinched.)
+Let $s_{Y,J}: X\cup_Y (Y\times J) \to X$ be a collaring homeomorphism
+(see the end of \S\ref{ss:syst-o-fields}).
+Here we use $Y\times J$ with boundary entirely pinched.
 We define a map
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
diff -r 27cfae8f4330 -r 3ae1a110873b text/tqftreview.tex
--- a/text/tqftreview.tex	Tue Mar 15 07:25:13 2011 -0700
+++ b/text/tqftreview.tex	Tue Mar 15 08:01:12 2011 -0700
@@ -214,17 +214,28 @@
 \medskip
 
 
-Using the functoriality and product field properties above, together
-with boundary collar homeomorphisms of manifolds, we can define 
-{\it collar maps} $\cC(M)\to \cC(M)$.
 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
 of $\bd M$.
+Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
+Extend the product structure on $Y\times I$ to a bicollar neighborhood of 
+$Y$ inside $M \cup (Y\times I)$.
+We call a homeomorphism
+\[
+	f: M \cup (Y\times I) \to M
+\]
+a {\it collaring homeomorphism} if $f$ is the identity outside of the bicollar
+and $f$ preserves the fibers of the bicollar.
+
+Using the functoriality and product field properties above, together
+with collaring homeomorphisms, we can define 
+{\it collar maps} $\cC(M)\to \cC(M)$.
+Let $M$ and $Y \sub \bd M$ be as above.
 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable 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 $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
 Then we call the map $x \mapsto f(x \bullet (c\times I))$ a {\it collar map}.
+
 We call the equivalence relation generated by collar maps and
 homeomorphisms isotopic to the identity {\it extended isotopy}, since the collar maps
 can be thought of (informally) as the limit of homeomorphisms