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205 close to the universal bi-Lipschitz constant for $f$. |
205 close to the universal bi-Lipschitz constant for $f$. |
206 |
206 |
207 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
207 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
208 \end{proof} |
208 \end{proof} |
209 |
209 |
210 \begin{lemma} |
210 \begin{lemma} \label{extension_lemma_c} |
211 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 or PL homeomorphisms. |
211 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 or PL homeomorphisms. |
212 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
212 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
213 of $X$. |
213 of $X$. |
214 Then $G_*$ is a strong deformation retract of $\cX_*$. |
214 Then $G_*$ is a strong deformation retract of $\cX_*$. |
215 \end{lemma} |
215 \end{lemma} |