4 

     5 

     6 \medskip

     7 \hrule

     8 \medskip

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

    10 in this section}

    11 

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

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

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

    15 \eq{

    16     P = P_1 \times \cdots \times P_m

    17 }

    18 (for some $m \le k$)

    19 and families of homeomorphisms

    20 \eq{

    21     f_i :  P_i \times X \to X

    22 }

    23 such that

    24 \begin{itemize}

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

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

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

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

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

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

    31 \end{itemize}

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

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

    34 \medskip

    35 \hrule

    36 \medskip

    37 \nn{another refugee:}

    38 

    39 We will actually prove the following more general result.

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

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

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

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

    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}

    48 

    49 \begin{lemma}  \label{extension_lemma_b}

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

    51 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$.

    52 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.

    53 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with

    54 chains of smooth maps or immersions.

    55 \end{lemma}

    56 

    57 \medskip

    58 \hrule

    59 \medskip

    60