37 \item If $f$ has support $S\sub X$, then |
37 \item If $f$ has support $S\sub X$, then |
38 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
38 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
39 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
39 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
40 has support $S' \subset X$, then |
40 has support $S' \subset X$, then |
41 $F: (I\times Q)\times X\to T$ also has support $S'$. |
41 $F: (I\times Q)\times X\to T$ also has support $S'$. |
42 \item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case). |
42 \item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and |
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43 let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), |
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44 bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case). |
43 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
45 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
44 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
46 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
45 \end{enumerate} |
47 \end{enumerate} |
46 \end{lemma} |
48 \end{lemma} |
47 |
49 |
126 \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha) |
128 \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha) |
127 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
129 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
128 \right) . |
130 \right) . |
129 \end{equation} |
131 \end{equation} |
130 |
132 |
131 This completes the definition of $u: I \times P \times X \to P$. The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, since one of the normal coordinates becomes $0$ or $1$. |
133 This completes the definition of $u: I \times P \times X \to P$. |
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134 The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and |
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135 a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, |
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136 since one of the normal coordinates becomes $0$ or $1$. |
132 Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$. |
137 Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$. |
133 |
138 |
134 \medskip |
139 \medskip |
135 |
140 |
136 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
141 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
206 |
211 |
207 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
212 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
208 \end{proof} |
213 \end{proof} |
209 |
214 |
210 \begin{lemma} \label{extension_lemma_c} |
215 \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. |
216 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the |
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217 subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, |
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218 bi-Lipschitz homeomorphisms or PL homeomorphisms. |
212 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
219 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
213 of $X$. |
220 of $X$. |
214 Then $G_*$ is a strong deformation retract of $\cX_*$. |
221 Then $G_*$ is a strong deformation retract of $\cX_*$. |
215 \end{lemma} |
222 \end{lemma} |
216 \begin{proof} |
223 \begin{proof} |