47 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
47 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
48 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
48 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
49 \end{enumerate} |
49 \end{enumerate} |
50 \end{lemma} |
50 \end{lemma} |
51 |
51 |
52 Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological |
52 %Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological |
53 homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below. |
53 %homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below. |
54 Since the proof is rather different we segregate it to a separate lemma. |
54 %Since the proof is rather different we segregate it to a separate lemma. |
55 |
55 |
56 \begin{proof} |
56 \begin{proof} |
57 Our homotopy will have the form |
57 Our homotopy will have the form |
58 \eqar{ |
58 \eqar{ |
59 F: I \times P \times X &\to& X \\ |
59 F: I \times P \times X &\to& X \\ |
219 \end{proof} |
219 \end{proof} |
220 |
220 |
221 |
221 |
222 % Edwards-Kirby: MR0283802 |
222 % Edwards-Kirby: MR0283802 |
223 |
223 |
|
224 \noop { %%%%%% begin \noop %%%%%%%%%%%%%%%%%%%%%%% |
|
225 |
224 The above proof doesn't work for homeomorphisms which are merely continuous. |
226 The above proof doesn't work for homeomorphisms which are merely continuous. |
225 The $k=1$ case for plain, continuous homeomorphisms |
227 The $k=1$ case for plain, continuous homeomorphisms |
226 is more or less equivalent to Corollary 1.3 of \cite{MR0283802}. |
228 is more or less equivalent to Corollary 1.3 of \cite{MR0283802}. |
227 The proof found in \cite{MR0283802} of that corollary can be adapted to many-parameter families of |
229 The proof found in \cite{MR0283802} of that corollary can be adapted to many-parameter families of |
228 homeomorphisms: |
230 homeomorphisms: |
344 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
346 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
345 and the subdivision mentioned above is chosen fine enough to insure this. |
347 and the subdivision mentioned above is chosen fine enough to insure this. |
346 |
348 |
347 \end{proof} |
349 \end{proof} |
348 |
350 |
349 |
351 } %%%%%% end \noop %%%%%%%%%%%%%%%%%%% |
350 |
352 |
351 \begin{lemma} \label{extension_lemma_c} |
353 \begin{lemma} \label{extension_lemma_c} |
352 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the |
354 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the |
353 subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, |
355 subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, |
354 bi-Lipschitz homeomorphisms, PL homeomorphisms or plain old continuous homeomorphisms. |
356 bi-Lipschitz homeomorphisms, or PL homeomorphisms. |
355 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
357 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
356 of $X$. |
358 of $X$. |
357 Then $G_*$ is a strong deformation retract of $\cX_*$. |
359 Then $G_*$ is a strong deformation retract of $\cX_*$. |
358 \end{lemma} |
360 \end{lemma} |
359 \begin{proof} |
361 \begin{proof} |
360 It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
362 It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
361 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
363 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
362 This is exactly what Lemma \ref{basic_adaptation_lemma} (or \ref{basic_adaptation_lemma_2}) |
364 This is exactly what Lemma \ref{basic_adaptation_lemma} |
363 gives us. |
365 gives us. |
364 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
366 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
365 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
367 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
366 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
368 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
367 Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$, |
369 Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$, |