
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




