%!TEX root = ../../blob1.tex

\section{Adapting families of maps to open covers}  \label{sec:localising}

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

\[

	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 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.

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{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}

\begin{proof}

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 makes the conclusions of the lemma true.

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.

\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 $\jj$.

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 $\jj$.

For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$

and let $p(D, \alpha) = p(C(D, \alpha))$.

For $p \in D$ we define

\eq{

    u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p(D, \alpha) .

}

(Recall that $P$ is a convex linear polyhedron, so the weighted average of points of $P$

makes sense.)

Thus far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $\jj$.

We will now extend $u$ inductively to handles of index less than $k$.

Let $E$ be a $k{-}1$-handle.

$E$ is homeomorphic to $B^{k-1}\times [0,1]$, and meets

the $k$-handles at $B^{k-1}\times\{0\}$ and $B^{k-1}\times\{1\}$.

Let $\eta : E \to [0,1]$, $\eta(x, s) = s$ be the normal coordinate

of $E$.

Let $D_0$ and $D_1$ be the two $k$-handles of $\jj$ adjacent to $E$.

There is at most one index $\beta$ such that $C(D_0, \beta) \ne C(D_1, \beta)$.

arbitrarily.)

For $p \in E$, define

\eq{

    u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p(D_0, \alpha)

            + r_\beta(x) (\eta(p) p(D_0, p) + (1-\eta(p)) p(D_1, p)) \right) .

}

Now define, for $p \in E$,

\begin{equation}

\label{eq:u}

    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) .

\end{equation}

This completes the definition of $u: I \times P \times X \to P$.

\medskip

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 \eqref{eq:u} 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}

\end{proof}

\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}

}

\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 $X$.

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}