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7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
8 unity $\{r_\alpha\}$. |
8 unity $\{r_\alpha\}$. |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
11 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) |
11 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) |
12 for all but finitely many $\alpha$. \nn{Can't we just assume $\cU$ is finite? Is something subtler happening? -S} |
12 for all but finitely many $\alpha$. |
13 |
13 |
14 Consider $C_*(\Maps(X\to T))$, the singular chains on the space of continuous maps from $X$ to $T$. |
14 Consider $C_*(\Maps(X\to T))$, the singular chains on the space of continuous maps from $X$ to $T$. |
15 $C_k(\Maps(X \to T))$ is generated by continuous maps |
15 $C_k(\Maps(X \to T))$ is generated by continuous maps |
16 \[ |
16 \[ |
17 f: P\times X \to T , |
17 f: P\times X \to T , |
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} |
216 \begin{proof} |
216 \begin{proof} |
217 If suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
217 It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
218 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
218 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
219 This is exactly what Lemma \ref{basic_adaptation_lemma} |
219 This is exactly what Lemma \ref{basic_adaptation_lemma} |
220 gives us. |
220 gives us. |
221 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
221 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
222 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
222 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |