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+%!TEX root = ../blob1.tex
+
+\section{Families of Diffeomorphisms}  \label{sec:localising}
+
+Lo, the proof of Lemma (\ref{extension_lemma}):
+
+\nn{should this be an appendix instead?}
+
+\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}
+
+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 \\
+    (t, p, x) &\mapsto& f(u(t, p, x), x)
+}
+for some function
+\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.
+
+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 $X\to X$.
+
+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 $\sum c_\alpha \le k$.
+
+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 $L$.
+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 $P$.
+
+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 $P$
+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
+interpolate between the values on $k$-handles defined above.
+
+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 $E$
+with a $k$-handle.
+Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
+corresponding to $E$.
+Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
+adjacent to the $k{-}1$-cell corresponding to $E$.
+For $p \in E$, define
+\eq{
+    u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
+            + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
+}
+
+In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
+$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
+The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
+If we triangulate $R$ (without introducing new vertices), we can linearly extend
+a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
+Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
+the $k{-}j$-cell corresponding to $E$.
+For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
+Now define, for $p \in E$,
+\eq{
+    u(t, p, x) = (1-t)p + t \left(
+            \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
+                + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
+             \right) .
+}
+Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
+mentioned above.
+
+This completes the definition of $u: I \times P \times X \to P$.
+
+\medskip
+
+Next we verify that $u$ has the desired properties.
+
+Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
+Therefore $F$ is a homotopy from $f$ to something.
+
+Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
+then $F$ is a homotopy through diffeomorphisms.
+We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
+We have
+\eq{
+%   \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
+    \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
+}
+Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
+\nn{bounded away from zero, or something like that}.
+(Recall that $X$ and $P$ are compact.)
+Also, $\pd{f}{p}$ is bounded.
+So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
+It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
+(which is bounded)
+and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
+These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
+This completes the proof that $F$ is a homotopy through diffeomorphisms.
+
+\medskip
+
+Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
+is a singular cell adapted to $\cU$.
+This will complete the proof of the lemma.
+\nn{except for boundary issues and the `$P$ is a cell' assumption}
+
+Let $j$ be the codimension of $D$.
+(Or rather, the codimension of its corresponding cell.  From now on we will not make a distinction
+between handle and corresponding cell.)
+Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
+where the $j_i$'s are the codimensions of the $K_\alpha$
+cells of codimension greater than 0 which intersect to form $D$.
+We will show that
+if the relevant $U_\alpha$'s are disjoint, then
+$F(1, \cdot, \cdot) : D\times X \to X$
+is a product of singular cells of dimensions $j_1, \ldots, j_m$.
+If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
+cells whose dimensions correspond to a partition of the $j_i$'s.
+We will consider some simple special cases first, then do the general case.
+
+First consider the case $j=0$ (and $m=0$).
+A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
+is independent of $p \in P$.
+So the corresponding map $D \to \Diff(X)$ is constant.
+
+Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
+Now Equation yyyy applies.
+We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
+It follows that the singular cell $D \to \Diff(X)$ can be written as a product
+of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
+The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
+
+Next case: $j=2$, $m=1$, $j_1 = 2$.
+This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
+1-dimensional.
+We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
+and a 2-cell with support $U_\beta$.
+
+Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
+In this case the codimension 2 cell $D$ is the intersection of two
+codimension 1 cells, from $K_\beta$ and $K_\gamma$.
+We can write $D = D' \times I \times I$, where the normal coordinates are constant
+on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
+If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
+two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
+If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
+a 2-cell supported on $U_\beta \cup U_\gamma$.
+\nn{need to check that this is true}
+
+\nn{finally, general case...}
+
+\nn{this completes proof}
+
+\input{text/explicit.tex}
+