--- a/text/evmap.tex Wed Apr 07 22:39:34 2010 -0700
+++ b/text/evmap.tex Sat Apr 10 18:03:36 2010 -0700
@@ -237,7 +237,8 @@
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
%We also have that $\deg(b'') = 0 = \deg(p'')$.
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
-This is possible by \ref{bcontract}, \ref{disjunion} and \nn{property 2 of local relations (isotopy)}.
+This is possible by \ref{bcontract}, \ref{disjunion} and the fact that isotopic fields
+differ by a local relation \nn{give reference?}.
Finally, define
\[
e(p\ot b) \deq x' \bullet p''(b'') .
@@ -337,7 +338,7 @@
\begin{proof}
Let $c$ be a subset of the blobs of $b$.
-There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$
+There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$
and all such $c$.
(Here we are using a piecewise smoothness assumption for $\bd c$, and also
the fact that $\bd c$ is collared.
@@ -351,7 +352,7 @@
Choose $k = k_{bmn}$ such that
\[
- t\ep_k < l
+ t\ep_k < \lambda
\]
and
\[
@@ -375,12 +376,17 @@
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b)
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) .
\]
+Recall that
+\[
+ N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|).
+\]
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$
of $|q|$, each homeomorphic to a disjoint union of balls, with
\[
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) .
\]
-The inequalities above \nn{give ref} guarantee that we can find $u_l$ with
+The inequalities above guarantee that
+for each $0\le l\le m$ we can find $u_l$ with
\[
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i
\]
@@ -452,7 +458,8 @@
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$.
For convenience, let $\phi_0 = 0$.
-Assume also that $\phi_k r \le \rho(M)$.
+Assume also that $\phi_k r \le \rho(M)$,
+where $\rho(M)$ is as in Lemma \ref{xxzz11}.
Then there exists a neighborhood $U$ of $S$,
homeomorphic to a disjoint union of balls, such that
\[