
diff -r 630ceb40a07b -r 7afacaa87bdb text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Thu May 27 14:15:19 2010 -0700
+++ b/text/appendixes/famodiff.tex	Thu May 27 15:06:48 2010 -0700
@@ -9,14 +9,10 @@
 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$;
 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.)
 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) 
-for all but finitely many $\alpha$.
+for all but finitely many $\alpha$. \nn{Can't we just assume $\cU$ is finite? Is something subtler happening? -S}
 
-Let
-\[
-	CM_*(X, T) \deq C_*(\Maps(X\to T)) ,
-\]
-the singular chains on the space of continuous maps from $X$ to $T$.
-$CM_k(X, T)$ is generated by continuous maps
+Consider  $C_*(\Maps(X\to T))$, the singular chains on the space of continuous maps from $X$ to $T$.
+$C_k(\Maps(X \to T))$ is generated by continuous maps
 \[
 	f: P\times X \to T ,
 \]
@@ -24,7 +20,7 @@
 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when
 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if 
 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s.
-A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of 
+A chain $c \in C_*(\Maps(X \to T))$ is adapted to $\cU$ if it is a linear combination of 
 generators which are adapted.
 
 \begin{lemma} \label{basic_adaptation_lemma}
@@ -40,14 +36,12 @@
 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$.
 \item If $f$ has support $S\sub X$, then
 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$.
-Furthermore, if $Q\sub\bd P$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$
-has support $S'$, then
+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 If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a 
-[immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism]
-then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$.
-(Of course we must assume that $X$ and $T$ are the appropriate 
-sort of manifolds for this to make sense.)
+\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}
 \end{lemma}
 
@@ -80,7 +74,7 @@
 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$.
 If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$.
 (It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$.
-This puts some mild constraints on the choice of $K_\alpha$.)
+Ensuring this is possible corresponds to some mild constraints on the choice of the $K_\alpha$.)
 
 Let $D$ be a $k$-handle of $\jj$.
 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$
@@ -134,7 +128,7 @@
              \right) .
 \end{equation}
 
-This completes the definition of $u: I \times P \times X \to P$.
+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
@@ -150,7 +144,7 @@
 Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$
 is a singular cell adapted to $\cU$.
 Let $k-j$ be the index of $D$.
-Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if 
+Referring to Equation \eqref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if 
 $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if
 $x\in \bigcup_{\beta\in\cN} U_\beta$.
 Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$,
@@ -176,7 +170,7 @@
 \medskip
 
 Now for claim 4 of the lemma.
-Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms.
+Assume that $X$ and $T$ are smooth manifolds and that $f$ is a smooth family of diffeomorphisms.
 We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a 
 diffeomorphism for all $t$ and $p$.
 It suffices to 
@@ -188,8 +182,8 @@
 }
 Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, 
 $\pd{f}{x}$ is non-singular and bounded away from zero.
-Also, $\pd{f}{p}$ is bounded.
-So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
+Also, since $f$ is smooth $\pd{f}{p}$ is bounded.
+Thus if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
 (which is bounded)
 and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s.
@@ -200,7 +194,7 @@
 through essentially unchanged.
 
 Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms.
-We assume that $f$ is Lipschitz in $P$ direction as well.
+Recall that we assume that $f$ is Lipschitz in the $P$ direction as well.
 The argument in this case is similar to the one above for diffeomorphisms, with
 bounded partial derivatives replaced by Lipschitz constants.
 Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for 
@@ -214,15 +208,14 @@
 \end{proof}
 
 \begin{lemma}
-Let $CM_*(X\to T)$ be the singular chains on the space of continuous maps
-[resp. immersions, diffeomorphisms, PL homeomorphisms, bi-Lipschitz homeomorphisms]
-from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$
+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 $CM_*(X\to T)$.
+Then $G_*$ is a strong deformation retract of $\cX_*$.
 \end{lemma}
 \begin{proof}
-If suffices to show that given a generator $f:P\times X\to T$ of $CM_k(X\to T)$ with
-$\bd f \in G_{k-1}$ there exists $h\in CM_{k+1}(X\to T)$ with $\bd h = f + g$ and $g \in G_k$.
+If suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with
+$\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$.
 This is exactly what Lemma \ref{basic_adaptation_lemma}
 gives us.
 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$.




