3 \section{Families of Diffeomorphisms}  \label{sec:localising}

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

     4 

     4 

     5 

     5 

     6 \medskip

     6 Let $X$ and $T$ be topological spaces.

     7 \hrule

     7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of

     8 \medskip

     8 unity $\{r_\alpha\}$.

     9 \nn{the following was removed from earlier section; it should be reincorporated somehwere

     9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$;

    10 in this section}

    10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.)

    11 

    11 

    12 Let $\cU = \{U_\alpha\}$ be an open cover of $X$.

    12 Let

    13 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is

    13 \[

    14 {\it adapted to $\cU$} if there is a factorization

    14 	CM_*(X, T) \deq C_*(\Maps(X\to T)) ,

    15 \eq{

    15 \]

    16     P = P_1 \times \cdots \times P_m

    16 the singular chains on the space of continuous maps from $X$ to $T$.

    17 }

    17 $CM_k(X, T)$ is generated by continuous maps

    18 (for some $m \le k$)

    18 \[

    19 and families of homeomorphisms

    19 	f: P\times X \to T ,

    20 \eq{

    20 \]

    21     f_i :  P_i \times X \to X

    21 where $P$ is some linear polyhedron in $\r^k$.

    22 }

    22 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when

    23 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if 

    24 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s.

    25 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of 

    26 generators which are adapted.

    27 

    28 \begin{lemma} \label{basic_adaptation_lemma}

    29 The $f: P\times X \to T$, as above.

    30 The there exists

    31 \[

    32 	F: I \times P\times X \to T

    33 \]

    24 \begin{itemize}

    35 \begin{enumerate}

    25 \item each $f_i$ is supported on some connected $V_i \sub X$;

    36 \item $F(0, \cdot, \cdot) = f$ .

    26 \item the sets $V_i$ are mutually disjoint;

    37 \item We can decompose $P = \cup_i D_i$ so that

    27 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,

    38 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$.

    28 where $k_i = \dim(P_i)$; and

    39 \item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction

    29 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$

    40 $F: (I\times Q)\times X\to T$ also has support $S$.

    30 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$.

    41 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a 

    31 \end{itemize}

    42 [submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism]

    32 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum

    43 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$.

    33 of singular cells, each of which is adapted to $\cU$.

    44 (Of course we must assume that $X$ and $T$ are the appropriate 

    34 \medskip

    45 sort of manifolds for this to make sense.)

    35 \hrule

    46 \end{enumerate}

    36 \medskip

    47 \end{lemma}

    37 \nn{another refugee:}

    48 

    38 

    49 

    39 We will actually prove the following more general result.

    50 

    40 Let $S$ and $T$ be an arbitrary topological spaces.

    51 

    41 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$;

    52 \noop{

    42 %check this; or maybe just restrict the cover}

    53 

    43 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps

    54 \nn{move this to later:}

    44 from $S$ to $T$.

    45 Let $\cU$ be an open cover of $S$ which affords a partition of unity.

    46 \nn{for some $S$ and $\cU$ there is no partition of unity?  like if $S$ is not paracompact?

    47 in any case, in our applications $S$ will always be a manifold}

    50 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$.

    57 Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$.