
diff -r 0bc6fa29b62a -r 24be062a87a1 text/evmap.tex
--- a/text/evmap.tex	Sun Sep 19 23:15:21 2010 -0700
+++ b/text/evmap.tex	Mon Sep 20 06:10:49 2010 -0700
@@ -80,7 +80,8 @@
 \end{lemma}
 
 \begin{proof}
-It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that 
+It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated 
+pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that 
 \[
 	(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))
 \]
@@ -156,7 +157,8 @@
 disjoint union of balls.
 
 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and
-also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
+also satisfying conditions specified below. 
+\nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
 As before, choose a sequence of collar maps $f_j$ 
 such that each has support
 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms
@@ -223,7 +225,8 @@
 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous.
 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous,
 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on
-$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
+$\bc_0(B)$ comes from the generating set $\BD_0(B)$. 
+\nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
 \end{itemize}
 
 We can summarize the above by saying that in the typical continuous family
@@ -270,7 +273,8 @@
 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$.
 
 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron.
-We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.}
+We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. 
+\nn{I found it pretty confusing to reuse the letter $r$ here.}
 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking
 the same value (namely $r(y(p))$, for any $p\in P$).
 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams
@@ -302,7 +306,10 @@
 			&= x - r(x) + r(x) \\
 			&= x.
 \end{align*}
-Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
+Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, 
+as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ 
+\nn{explain why this is true?} 
+and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
 
 For $x\in \btc_{00}$ we have
 \begin{align*}
@@ -517,10 +524,12 @@
 \end{equation*}
 \end{enumerate}
 Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ 
-satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy.
+satisfying the above two conditions which is $m$-connected. In particular, 
+this means that the choice of chain map above is unique up to homotopy.
 \end{thm}
 \begin{rem}
-Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this.
+Note that the statement doesn't quite give uniqueness up to iterated homotopy. 
+We fully expect that this should actually be the case, but haven't been able to prove this.
 \end{rem}
 
 
@@ -712,8 +721,8 @@
 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 Properties \ref{property:disjoint-union} and \ref{property:contractibility}  and the fact that isotopic fields
-differ by a local relation.
+This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} 
+and the fact that isotopic fields differ by a local relation.
 Finally, define
 \[
 	e(p\ot b) \deq x' \bullet p''(b'') .
@@ -829,7 +838,8 @@
 
 \begin{proof}
 
-There exists $\lambda > 0$ such that for every  subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
+There exists $\lambda > 0$ such that for every  subset $c$ of the blobs of $b$ the set 
+$\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
 (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




