--- a/text/evmap.tex Mon Jul 12 17:29:25 2010 -0600
+++ b/text/evmap.tex Mon Jul 12 21:08:14 2010 -0600
@@ -122,7 +122,7 @@
Now for a little more detail.
(But we're still just motivating the full, gory details, which will follow.)
-Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$.
+Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$.
By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families
$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls.
For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough
@@ -153,7 +153,7 @@
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
Choose a metric on $X$.
-Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero
+Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
(e.g.\ $\ep_i = 2^{-i}$).
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
@@ -177,7 +177,7 @@
is homeomorphic to a disjoint union of balls and
\[
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b)
- \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) .
+ \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) ,
\]
and further $\bd(p\ot b) \in G_*^{i,m}$.
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
@@ -345,7 +345,8 @@
\begin{proof}
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
-(Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.)
+(Here we are using the fact that the blobs are
+piecewise smooth or piecewise-linear and that $\bd c$ is collared.)
We need to consider all such $c$ because all generators appearing in
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.)