
diff -r cb40431c8a65 -r a7a23eeb5d65 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Sun May 16 17:15:00 2010 -0700
+++ b/text/appendixes/famodiff.tex	Tue May 18 22:49:17 2010 -0600
@@ -3,11 +3,14 @@
 \section{Adapting families of maps to open covers}  \label{sec:localising}
 
 
-Let $X$ and $T$ be topological spaces.
+Let $X$ and $T$ be topological spaces, with $X$ compact.
 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 $\sum_\alpha r_\alpha = 1$.)
+Since $X$ is compact, we will further assume that $r_\alpha \ne 0$ (globally) 
+for only finitely
+many $\alpha$.
 
 Let
 \[
@@ -18,7 +21,7 @@
 \[
 	f: P\times X \to T ,
 \]
-where $P$ is some linear polyhedron in $\r^k$.
+where $P$ is some convex 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.
@@ -48,61 +51,7 @@
 
 
 
-
-\noop{
-
-\nn{move this to later:}
-
-\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{**** resume revising here ****}
-
-
 \begin{proof}
-
-Recall that we are given
-an open cover $\cU = \{U_\alpha\}$ and an
-$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
-We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
-
-Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
-
-As a first approximation to the argument we will eventually make, let's replace $x$
-with a single singular cell
-\eq{
-    f: P \times X \to X .
-}
-Also, we'll ignore for now issues around $\bd P$.
-
 Our homotopy will have the form
 \eqar{
     F: I \times P \times X &\to& X \\
@@ -112,42 +61,40 @@
 \eq{
     u : I \times P \times X \to P .
 }
-First we describe $u$, then we argue that it does what we want it to do.
+
+First we describe $u$, then we argue that it makes the conclusions of the lemma true.
 
-For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
-The various $K_\alpha$ should be in general position with respect to each other.
-We will see below that the $K_\alpha$'s need to be sufficiently fine in order
-to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
-merely a homotopy through maps $X\to X$.
+For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$
+such that the various $K_\alpha$ are in general position with respect to each other.
+If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$
+sufficiently fine as described below.
 
-Let $L$ be the union of all the $K_\alpha$'s.
-$L$ is itself a cell decomposition of $P$.
-\nn{next two sentences not needed?}
-To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
-where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
-Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
+\def\jj{\tilde{L}}
+Let $L$ be a common refinement all the $K_\alpha$'s.
+Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$.
+Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and,
+more importantly for our purposes, a $k{-}i$-dimensional normal coordinate.
+We will typically use the same notation for $i$-cells of $L$ and the 
+corresponding $i$-handles of $\jj$.
 
-Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
-Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
-more importantly, a $k{-}i$-dimensional normal coordinate.
-
-For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
-Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding
-$k$-cell of $L$.
+For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p_c \in C \sub P$.
+Let $D$ be a $k$-handle of $\jj$.
 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
-which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
+which contain $D$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
 
 For $p \in D$ we define
 \eq{
     u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
 }
-(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
+(Recall that $P$ is a convex linear polyhedron, so the weighted average of points of $P$
 makes sense.)
 
-So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
-For handles of $J$ of index less than $k$, we will define $u$ to
+So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $\jj$.
+For handles of $\jj$ of index less than $k$, we will define $u$ to
 interpolate between the values on $k$-handles defined above.
 
+\nn{*** resume revising here ***}
+
 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
 of $E$.
 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
@@ -268,10 +215,47 @@
 
 
 
+\noop{
+
+\nn{move this to later:}
+
+\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
-\nn{the following was removed from earlier section; it should be reincorporated somehwere
+
+
+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}
+
+}
+
+
+
+
+\medskip
+\hrule
+\medskip
+\nn{the following was removed from earlier section; it should be reincorporated somewhere
 in this section}
 
 Let $\cU = \{U_\alpha\}$ be an open cover of $X$.




