
diff -r 52309e058a95 -r 2252c53bd449 text/evmap.tex
--- a/text/evmap.tex	Sat May 29 23:13:03 2010 -0700
+++ b/text/evmap.tex	Sat May 29 23:13:20 2010 -0700
@@ -41,7 +41,8 @@
 I lean toward the latter.}
 \medskip
 
-The proof will occupy the the next several pages.
+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.
+
 Without loss of generality, we will assume $X = Y$.
 
 \medskip
@@ -108,7 +109,7 @@
 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in
 this case a 0-blob diagram).
 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ 
-(by \ref{disjunion} and \ref{bcontract}).
+(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}).
 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
 such that 
@@ -153,8 +154,7 @@
 
 \medskip
 
-Now for the details.
-
+\begin{proof}[Proof of Proposition \ref{CHprop}.]
 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
 
 Choose a metric on $X$.
@@ -313,7 +313,7 @@
 $G_*^{i,m}$.
 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
-Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
+Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{extension_lemma}.
 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
 supports.
 Define
@@ -610,26 +610,10 @@
 \end{itemize}
 
 
-\nn{to be continued....}
-
-\noop{
-
-\begin{lemma}
-
-\end{lemma}
-
-\begin{proof}
-
 \end{proof}
 
-}
+\nn{to be continued....}
 
 
 
 
-%\nn{say something about associativity here}
-
-
-
-
-




