--- a/text/evmap.tex Tue Apr 06 22:39:49 2010 -0700
+++ b/text/evmap.tex Wed Apr 07 22:39:34 2010 -0700
@@ -176,7 +176,8 @@
\[
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).
\]
-In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
+In other words, for each $i$
+we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling
the size of the buffers around $|p|$.
@@ -225,7 +226,7 @@
(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)
We therefore have splittings
\[
- p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' ,
+ p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' ,
\]
where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$,
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$,
@@ -315,7 +316,7 @@
$G_*^{i,m}$.
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 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{xxxxx}.
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
supports.
Define
@@ -324,8 +325,8 @@
\]
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$
-(depending on $b$, $n = \deg(p)$ and $m$).
-(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.)
+(depending on $b$, $\deg(p)$ and $m$).
+%(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.)
\begin{lemma} \label{Gim_approx}
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$.
@@ -341,7 +342,7 @@
(Here we are using a piecewise smoothness assumption for $\bd c$, and also
the fact that $\bd c$ is collared.
We need to consider all such $c$ because all generators appearing in
-iterated boundaries of must be in $G_*^{i,m}$.)
+iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.)
Let $r = \deg(b)$ and
\[