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 

   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}