1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
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3 \section{Adapting families of maps to open covers} \label{sec:localising} |
3 \section{Adapting families of maps to open covers} \label{sec:localising} |
4 |
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5 |
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6 Let $X$ and $T$ be topological spaces. |
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 $\sum_\alpha r_\alpha = 1$.) |
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11 Since $X$ is compact, we will further assume that $r_\alpha \ne 0$ (globally) |
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12 for only finitely |
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13 many $\alpha$. |
11 |
14 |
12 Let |
15 Let |
13 \[ |
16 \[ |
14 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
17 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
15 \] |
18 \] |
16 the singular chains on the space of continuous maps from $X$ to $T$. |
19 the singular chains on the space of continuous maps from $X$ to $T$. |
17 $CM_k(X, T)$ is generated by continuous maps |
20 $CM_k(X, T)$ is generated by continuous maps |
18 \[ |
21 \[ |
19 f: P\times X \to T , |
22 f: P\times X \to T , |
20 \] |
23 \] |
21 where $P$ is some linear polyhedron in $\r^k$. |
24 where $P$ is some convex linear polyhedron in $\r^k$. |
22 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
25 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
23 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
26 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
24 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
27 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
25 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
28 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
26 generators which are adapted. |
29 generators which are adapted. |
46 \end{enumerate} |
49 \end{enumerate} |
47 \end{lemma} |
50 \end{lemma} |
48 |
51 |
49 |
52 |
50 |
53 |
51 |
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52 \noop{ |
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53 |
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54 \nn{move this to later:} |
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55 |
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56 \begin{lemma} \label{extension_lemma_b} |
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57 Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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58 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
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59 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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60 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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61 chains of smooth maps or immersions. |
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62 \end{lemma} |
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63 |
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64 \medskip |
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65 \hrule |
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66 \medskip |
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67 |
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68 |
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69 In this appendix we provide the proof of |
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70 \nn{should change this to the more general \ref{extension_lemma_b}} |
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71 |
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72 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
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73 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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74 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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75 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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76 \end{lem*} |
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77 |
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78 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
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79 later draft} |
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80 |
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81 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
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82 about boundary later} |
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83 |
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84 } |
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85 |
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86 |
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87 \nn{**** resume revising here ****} |
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88 |
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89 |
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90 \begin{proof} |
54 \begin{proof} |
91 |
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92 Recall that we are given |
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93 an open cover $\cU = \{U_\alpha\}$ and an |
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94 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
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95 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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96 |
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97 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
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98 |
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99 As a first approximation to the argument we will eventually make, let's replace $x$ |
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100 with a single singular cell |
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101 \eq{ |
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102 f: P \times X \to X . |
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103 } |
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104 Also, we'll ignore for now issues around $\bd P$. |
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105 |
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106 Our homotopy will have the form |
55 Our homotopy will have the form |
107 \eqar{ |
56 \eqar{ |
108 F: I \times P \times X &\to& X \\ |
57 F: I \times P \times X &\to& X \\ |
109 (t, p, x) &\mapsto& f(u(t, p, x), x) |
58 (t, p, x) &\mapsto& f(u(t, p, x), x) |
110 } |
59 } |
111 for some function |
60 for some function |
112 \eq{ |
61 \eq{ |
113 u : I \times P \times X \to P . |
62 u : I \times P \times X \to P . |
114 } |
63 } |
115 First we describe $u$, then we argue that it does what we want it to do. |
64 |
116 |
65 First we describe $u$, then we argue that it makes the conclusions of the lemma true. |
117 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
66 |
118 The various $K_\alpha$ should be in general position with respect to each other. |
67 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$ |
119 We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
68 such that the various $K_\alpha$ are in general position with respect to each other. |
120 to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
69 If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$ |
121 merely a homotopy through maps $X\to X$. |
70 sufficiently fine as described below. |
122 |
71 |
123 Let $L$ be the union of all the $K_\alpha$'s. |
72 \def\jj{\tilde{L}} |
124 $L$ is itself a cell decomposition of $P$. |
73 Let $L$ be a common refinement all the $K_\alpha$'s. |
125 \nn{next two sentences not needed?} |
74 Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$. |
126 To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
75 Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and, |
127 where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
76 more importantly for our purposes, a $k{-}i$-dimensional normal coordinate. |
128 Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
77 We will typically use the same notation for $i$-cells of $L$ and the |
129 |
78 corresponding $i$-handles of $\jj$. |
130 Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
79 |
131 Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
80 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p_c \in C \sub P$. |
132 more importantly, a $k{-}i$-dimensional normal coordinate. |
81 Let $D$ be a $k$-handle of $\jj$. |
133 |
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134 For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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135 Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
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136 $k$-cell of $L$. |
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137 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
82 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
138 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
83 which contain $D$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
139 |
84 |
140 For $p \in D$ we define |
85 For $p \in D$ we define |
141 \eq{ |
86 \eq{ |
142 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
87 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
143 } |
88 } |
144 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
89 (Recall that $P$ is a convex linear polyhedron, so the weighted average of points of $P$ |
145 makes sense.) |
90 makes sense.) |
146 |
91 |
147 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
92 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $\jj$. |
148 For handles of $J$ of index less than $k$, we will define $u$ to |
93 For handles of $\jj$ of index less than $k$, we will define $u$ to |
149 interpolate between the values on $k$-handles defined above. |
94 interpolate between the values on $k$-handles defined above. |
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95 |
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96 \nn{*** resume revising here ***} |
150 |
97 |
151 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
98 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
152 of $E$. |
99 of $E$. |
153 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
100 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
154 with a $k$-handle. |
101 with a $k$-handle. |
266 \end{proof} |
213 \end{proof} |
267 |
214 |
268 |
215 |
269 |
216 |
270 |
217 |
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218 \noop{ |
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219 |
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220 \nn{move this to later:} |
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221 |
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222 \begin{lemma} \label{extension_lemma_b} |
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223 Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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224 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
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225 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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226 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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227 chains of smooth maps or immersions. |
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228 \end{lemma} |
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229 |
271 \medskip |
230 \medskip |
272 \hrule |
231 \hrule |
273 \medskip |
232 \medskip |
274 \nn{the following was removed from earlier section; it should be reincorporated somehwere |
233 |
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234 |
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235 In this appendix we provide the proof of |
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236 \nn{should change this to the more general \ref{extension_lemma_b}} |
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237 |
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238 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
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239 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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240 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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241 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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242 \end{lem*} |
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243 |
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244 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
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245 later draft} |
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246 |
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247 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
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248 about boundary later} |
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249 |
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250 } |
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251 |
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252 |
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253 |
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254 |
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255 \medskip |
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256 \hrule |
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257 \medskip |
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258 \nn{the following was removed from earlier section; it should be reincorporated somewhere |
275 in this section} |
259 in this section} |
276 |
260 |
277 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
261 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
278 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
262 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
279 {\it adapted to $\cU$} if there is a factorization |
263 {\it adapted to $\cU$} if there is a factorization |