# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1274055300 25200
# Node ID cb40431c8a65ba785abffc0a978b4129c6d12057
# Parent  80c8e5d2f02b31672373792e7546ce968ecd725f
begin to revise families of maps appendix

diff -r 80c8e5d2f02b -r cb40431c8a65 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Sat May 15 17:21:59 2010 -0700
+++ b/text/appendixes/famodiff.tex	Sun May 16 17:15:00 2010 -0700
@@ -1,53 +1,60 @@
 %!TEX root = ../../blob1.tex
 
-\section{Families of Diffeomorphisms}  \label{sec:localising}
+\section{Adapting families of maps to open covers}  \label{sec:localising}
 
 
-\medskip
-\hrule
-\medskip
-\nn{the following was removed from earlier section; it should be reincorporated somehwere
-in this section}
+Let $X$ and $T$ be topological spaces.
+Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of
+unity $\{r_\alpha\}$.
+(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 $\sum_\alpha r_\alpha = 1$.)
 
-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
-\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
-}
+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
+\[
+	f: P\times X \to T ,
+\]
+where $P$ is some linear polyhedron in $\r^k$.
+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 
+generators which are adapted.
+
+\begin{lemma} \label{basic_adaptation_lemma}
+The $f: P\times X \to T$, as above.
+The there exists
+\[
+	F: I \times P\times X \to T
+\]
 such that
-\begin{itemize}
-\item each $f_i$ is supported on some connected $V_i \sub X$;
-\item the sets $V_i$ are mutually disjoint;
-\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 $g \in \Homeo(X)$.
-\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{another refugee:}
+\begin{enumerate}
+\item $F(0, \cdot, \cdot) = f$ .
+\item We can decompose $P = \cup_i D_i$ so that
+the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$.
+\item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction
+$F: (I\times Q)\times X\to T$ also has support $S$.
+\item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a 
+[submersion, 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.)
+\end{enumerate}
+\end{lemma}
 
-We will actually prove the following more general result.
-Let $S$ and $T$ be an arbitrary topological spaces.
-%\nn{might need to restrict $S$; the proof uses partition of unity on $S$;
-%check this; or maybe just restrict the cover}
-Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps
-from $S$ to $T$.
-Let $\cU$ be an open cover of $S$ which affords a partition of unity.
-\nn{for some $S$ and $\cU$ there is no partition of unity?  like if $S$ is not paracompact?
-in any case, in our applications $S$ will always be a manifold}
+
+
+
+\noop{
+
+\nn{move this to later:}
 
 \begin{lemma}  \label{extension_lemma_b}
-Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
+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
@@ -74,6 +81,12 @@
 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
 about boundary later}
 
+}
+
+
+\nn{**** resume revising here ****}
+
+
 \begin{proof}
 
 Recall that we are given
@@ -252,5 +265,44 @@
 
 \end{proof}
 
+
+
+
+\medskip
+\hrule
+\medskip
+\nn{the following was removed from earlier section; it should be reincorporated somehwere
+in this section}
+
+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
+\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
+}
+such that
+\begin{itemize}
+\item each $f_i$ is supported on some connected $V_i \sub X$;
+\item the sets $V_i$ are mutually disjoint;
+\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 $g \in \Homeo(X)$.
+\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
+
+
+
+
+
 \input{text/appendixes/explicit.tex}