
diff -r 966a571daa10 -r adfffac7c138 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Wed Jun 01 15:17:39 2011 -0600
+++ b/text/appendixes/famodiff.tex	Tue Jun 14 19:28:48 2011 -0600
@@ -47,6 +47,10 @@
 \end{enumerate}
 \end{lemma}
 
+Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological
+homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below.
+Since the proof is rather different we segregate it to a separate lemma.
+
 \begin{proof}
 Our homotopy will have the form
 \eqar{
@@ -212,10 +216,44 @@
 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well.
 \end{proof}
 
+
+% Edwards-Kirby: MR0283802
+
+The above proof doesn't work for homeomorphisms which are merely continuous.
+The $k=1$ case for plain, continuous homeomorphisms 
+is more or less equivalent to Corollary 1.3 of \cite{MR0283802}.
+The proof found in \cite{MR0283802} of that corollary can be adapted to many-parameter families of
+homeomorphisms:
+
+\begin{lemma} \label{basic_adaptation_lemma_2}
+Lemma \ref{basic_adaptation_lemma} holds for continuous homeomorphisms
+in item 4.
+\end{lemma}
+
+\begin{proof}
+We will imitate the proof of Corollary 1.3 of \cite{MR0283802}.
+
+Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$.
+After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$
+such that $g\circ f(P)$ is a small neighborhood of the 
+identity in $\Homeo(X)$.
+The sense of ``small" we mean will be explained below.
+It depends only on $\cU$ and some auxiliary covers.
+
+We assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+
+
+
+\nn{...}
+
+\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.
+bi-Lipschitz homeomorphisms, PL homeomorphisms or plain old continuous homeomorphisms.
 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$
 of $X$.
 Then $G_*$ is a strong deformation retract of $\cX_*$.
@@ -223,7 +261,7 @@
 \begin{proof}
 It 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}
+This is exactly what Lemma \ref{basic_adaptation_lemma} (or \ref{basic_adaptation_lemma_2})
 gives us.
 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$.
 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}.
@@ -234,6 +272,7 @@
 
 \medskip
 
+
 %%%%%% Lo, \noop{...}
 \noop{
 




