
diff -r 81d7c550b3da -r 7a67f45e2475 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Sat May 22 12:17:23 2010 -0600
+++ b/text/appendixes/famodiff.tex	Tue May 25 07:20:16 2010 -0700
@@ -199,62 +199,48 @@
 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well.
 \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 $\cU$
+of $X$.
+Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$.
+\end{lemma}
+\begin{proof}
+\nn{my current idea is too messy, so I'm going to wait and hopefully think
+of a cleaner proof}
 \noop{
-
-\nn{move this to later:}
+If suffices to show that 
+...
+Lemma \ref{basic_adaptation_lemma}
+...
+}
+\end{proof}
 
-\begin{lemma}  \label{extension_lemma_b}
-Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
-Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$.
-Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
-If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with
-chains of smooth maps or immersions.
-\end{lemma}
-
-\medskip
-\hrule
 \medskip
 
-
-In this appendix we provide the proof of
-\nn{should change this to the more general \ref{extension_lemma_b}}
-
-\begin{lem*}[Restatement of Lemma \ref{extension_lemma}]
-Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
-Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
-Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
-\end{lem*}
-
-\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
-later draft}
-
-\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
-about boundary later}
-
-}
-
-
-
+\nn{need to clean up references from the main text to the lemmas of this section}
 
 \medskip
-\hrule
-\medskip
-\nn{the following was removed from earlier section; it should be reincorporated somewhere
-in this section}
+
+\nn{do we want to keep the following?}
 
-Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
-A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is
-{\it adapted to $\cU$} if there is a factorization
+The above lemmas remain true if we replace ``adapted" with ``strongly adapted", as defined below.
+The proof of Lemma \ref{basic_adaptation_lemma} is modified by
+choosing the common refinement $L$ and interpolating maps $\eta$
+slightly more carefully.
+Since we don't need the stronger result, we omit the details.
+
+Let $X$, $T$ and $\cU$ be as above.
+A $k$-parameter family of maps $f: P \times X \to T$ is
+{\it strongly adapted to $\cU$} if there is a factorization
 \eq{
     P = P_1 \times \cdots \times P_m
 }
 (for some $m \le k$)
 and families of homeomorphisms
 \eq{
-    f_i :  P_i \times X \to X
+    f_i :  P_i \times X \to T
 }
 such that
 \begin{itemize}
@@ -263,17 +249,15 @@
 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
 where $k_i = \dim(P_i)$; and
 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
-for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$.
+for all $p = (p_1, \ldots, p_m)$, for some fixed $gX\to T$.
 \end{itemize}
-A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
-of singular cells, each of which is adapted to $\cU$.
+
+
 \medskip
 \hrule
 \medskip
 
-
-
-
+\nn{do we want to keep this alternative construction?}
 
 \input{text/appendixes/explicit.tex}
 




