diff -r 52e6b2d78dc4 -r 01c1daa71437 text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Sun Sep 25 22:13:07 2011 -0600 +++ b/text/appendixes/famodiff.tex Sun Sep 25 22:31:22 2011 -0600 @@ -49,9 +49,9 @@ \end{enumerate} \end{lemma} -Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological -homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below. -Since the proof is rather different we segregate it to a separate lemma. +%Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological +%homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below. +%Since the proof is rather different we segregate it to a separate lemma. \begin{proof} Our homotopy will have the form @@ -221,6 +221,8 @@ % Edwards-Kirby: MR0283802 +\noop { %%%%%% begin \noop %%%%%%%%%%%%%%%%%%%%%%% + The above proof doesn't work for homeomorphisms which are merely continuous. The $k=1$ case for plain, continuous homeomorphisms is more or less equivalent to Corollary 1.3 of \cite{MR0283802}. @@ -346,12 +348,12 @@ \end{proof} - +} %%%%%% end \noop %%%%%%%%%%%%%%%%%%% \begin{lemma} \label{extension_lemma_c} Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, -bi-Lipschitz homeomorphisms, PL homeomorphisms or plain old continuous homeomorphisms. +bi-Lipschitz homeomorphisms, or PL homeomorphisms. Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ of $X$. Then $G_*$ is a strong deformation retract of $\cX_*$. @@ -359,7 +361,7 @@ \begin{proof} It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. -This is exactly what Lemma \ref{basic_adaptation_lemma} (or \ref{basic_adaptation_lemma_2}) +This is exactly what Lemma \ref{basic_adaptation_lemma} gives us. More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}.