
diff -r 4718e0696bc6 -r c27e875508fd text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Fri Jun 04 20:37:38 2010 -0700
+++ b/text/appendixes/famodiff.tex	Fri Jun 04 20:43:14 2010 -0700
@@ -39,7 +39,9 @@
 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$
 has support $S' \subset X$, then
 $F: (I\times Q)\times X\to T$ also has support $S'$.
-\item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case).
+\item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and 
+let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), 
+bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case).
  If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$
 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$.
 \end{enumerate}
@@ -128,7 +130,10 @@
              \right) .
 \end{equation}
 
-This completes the definition of $u: I \times P \times X \to P$. The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, since one of the normal coordinates becomes $0$ or $1$. 
+This completes the definition of $u: I \times P \times X \to P$. 
+The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and 
+a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, 
+since one of the normal coordinates becomes $0$ or $1$. 
 Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$.
 
 \medskip
@@ -208,7 +213,9 @@
 \end{proof}
 
 \begin{lemma} \label{extension_lemma_c}
-Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms.
+Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the 
+subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, 
+bi-Lipschitz homeomorphisms or PL homeomorphisms.
 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$
 of $X$.
 Then $G_*$ is a strong deformation retract of $\cX_*$.




