6 Let $X$ and $T$ be topological spaces, with $X$ compact. |
6 Let $X$ and $T$ be topological spaces, with $X$ compact. |
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 $r_\alpha \ne 0$ (globally) |
11 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) |
12 for only finitely |
12 for all but finitely many $\alpha$. |
13 many $\alpha$. |
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14 |
13 |
15 Let |
14 Let |
16 \[ |
15 \[ |
17 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
16 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
18 \] |
17 \] |
27 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
26 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
28 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
27 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
29 generators which are adapted. |
28 generators which are adapted. |
30 |
29 |
31 \begin{lemma} \label{basic_adaptation_lemma} |
30 \begin{lemma} \label{basic_adaptation_lemma} |
32 The $f: P\times X \to T$, as above. |
31 Let $f: P\times X \to T$, as above. |
33 The there exists |
32 Then there exists |
34 \[ |
33 \[ |
35 F: I \times P\times X \to T |
34 F: I \times P\times X \to T |
36 \] |
35 \] |
37 such that |
36 such that |
38 \begin{enumerate} |
37 \begin{enumerate} |
39 \item $F(0, \cdot, \cdot) = f$ . |
38 \item $F(0, \cdot, \cdot) = f$ . |
40 \item We can decompose $P = \cup_i D_i$ so that |
39 \item We can decompose $P = \cup_i D_i$ so that |
41 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
40 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
42 \item If $f$ has support $S\sub X$, then |
41 \item If $f$ has support $S\sub X$, then |
43 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
42 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
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43 Furthermore, if $Q\sub\bd P$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
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44 has support $S'$, then |
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45 $F: (I\times Q)\times X\to T$ also has support $S'$. |
44 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
46 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
45 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
47 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
46 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
48 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
47 (Of course we must assume that $X$ and $T$ are the appropriate |
49 (Of course we must assume that $X$ and $T$ are the appropriate |
48 sort of manifolds for this to make sense.) |
50 sort of manifolds for this to make sense.) |
66 such that the various $K_\alpha$ are in general position with respect to each other. |
68 such that the various $K_\alpha$ are in general position with respect to each other. |
67 If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$ |
69 If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$ |
68 sufficiently fine as described below. |
70 sufficiently fine as described below. |
69 |
71 |
70 \def\jj{\tilde{L}} |
72 \def\jj{\tilde{L}} |
71 Let $L$ be a common refinement all the $K_\alpha$'s. |
73 Let $L$ be a common refinement of all the $K_\alpha$'s. |
72 Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$. |
74 Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$. |
73 Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and, |
75 Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and, |
74 more importantly for our purposes, a $k{-}i$-dimensional normal coordinate. |
76 more importantly for our purposes, a $k{-}i$-dimensional normal coordinate. |
75 We will typically use the same notation for $i$-cells of $L$ and the |
77 We will typically use the same notation for $i$-cells of $L$ and the |
76 corresponding $i$-handles of $\jj$. |
78 corresponding $i$-handles of $\jj$. |
77 |
79 |
78 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$. |
80 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$. |
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81 If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$. |
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82 (It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$. |
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83 This puts some mild constraints on the choice of $K_\alpha$.) |
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84 |
79 Let $D$ be a $k$-handle of $\jj$. |
85 Let $D$ be a $k$-handle of $\jj$. |
80 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ |
86 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ |
81 and let $p(D, \alpha) = p(C(D, \alpha))$. |
87 and let $p(D, \alpha) = p(C(D, \alpha))$. |
82 |
88 |
83 For $p \in D$ we define |
89 For $p \in D$ we define |
156 then |
163 then |
157 \[ |
164 \[ |
158 F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x) |
165 F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x) |
159 \] |
166 \] |
160 for all $(t,p)$ and $(t',p')$ in $I\times P$. |
167 for all $(t,p)$ and $(t',p')$ in $I\times P$. |
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168 Similarly, if $f(q,x) = f(q',x)$ for all $q,q'\in Q\sub \bd P$, |
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169 then |
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170 \[ |
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171 F(t, q, x) = f(u(t,q,x),x) = f(u(t',q',x),x) = F(t',q',x) |
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172 \] |
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173 for all $(t,q)$ and $(t',q')$ in $I\times Q$. |
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174 (Recall that we arranged above that $u(I\times Q\times X) \sub Q$.) |
161 |
175 |
162 \medskip |
176 \medskip |
163 |
177 |
164 Now for claim 4 of the lemma. |
178 Now for claim 4 of the lemma. |
165 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. |
179 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. |
205 from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$ |
219 from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$ |
206 of $X$. |
220 of $X$. |
207 Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$. |
221 Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$. |
208 \end{lemma} |
222 \end{lemma} |
209 \begin{proof} |
223 \begin{proof} |
210 \nn{my current idea is too messy, so I'm going to wait and hopefully think |
224 If suffices to show that given a generator $f:P\times X\to T$ of $CM_k(X\to T)$ with |
211 of a cleaner proof} |
225 $\bd f \in G_{k-1}$ there exists $h\in CM_{k+1}(X\to T)$ with $\bd h = f + g$ and $g \in G_k$. |
212 \noop{ |
226 This is exactly what Lemma \ref{basic_adaptation_lemma} |
213 If suffices to show that |
227 gives us. |
214 ... |
228 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
215 Lemma \ref{basic_adaptation_lemma} |
229 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
216 ... |
230 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
217 } |
231 Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$, |
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232 while part 3 of Lemma \ref{basic_adaptation_lemma} says that the restrictions to $I\times Q_i$ are in $G_k$. |
218 \end{proof} |
233 \end{proof} |
219 |
234 |
220 \medskip |
235 \medskip |
221 |
236 |
222 \nn{need to clean up references from the main text to the lemmas of this section} |
237 \nn{need to clean up references from the main text to the lemmas of this section} |
223 |
238 |
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239 %%%%%% Lo, \noop{...} |
224 \noop{ |
240 \noop{ |
225 |
241 |
226 \medskip |
242 \medskip |
227 |
243 |
228 \nn{do we want to keep the following?} |
244 \nn{do we want to keep the following?} |