1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
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2 |
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
3 \section{Adapting families of maps to open covers} \label{sec:localising} |
4 |
4 |
5 |
5 |
6 \medskip |
6 Let $X$ and $T$ be topological spaces. |
7 \hrule |
7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
8 \medskip |
8 unity $\{r_\alpha\}$. |
9 \nn{the following was removed from earlier section; it should be reincorporated somehwere |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
10 in this section} |
10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
11 |
11 |
12 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
12 Let |
13 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
13 \[ |
14 {\it adapted to $\cU$} if there is a factorization |
14 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
15 \eq{ |
15 \] |
16 P = P_1 \times \cdots \times P_m |
16 the singular chains on the space of continuous maps from $X$ to $T$. |
17 } |
17 $CM_k(X, T)$ is generated by continuous maps |
18 (for some $m \le k$) |
18 \[ |
19 and families of homeomorphisms |
19 f: P\times X \to T , |
20 \eq{ |
20 \] |
21 f_i : P_i \times X \to X |
21 where $P$ is some linear polyhedron in $\r^k$. |
22 } |
22 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
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23 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
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24 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
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25 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
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26 generators which are adapted. |
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27 |
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28 \begin{lemma} \label{basic_adaptation_lemma} |
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29 The $f: P\times X \to T$, as above. |
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30 The there exists |
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31 \[ |
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32 F: I \times P\times X \to T |
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33 \] |
23 such that |
34 such that |
24 \begin{itemize} |
35 \begin{enumerate} |
25 \item each $f_i$ is supported on some connected $V_i \sub X$; |
36 \item $F(0, \cdot, \cdot) = f$ . |
26 \item the sets $V_i$ are mutually disjoint; |
37 \item We can decompose $P = \cup_i D_i$ so that |
27 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
38 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
28 where $k_i = \dim(P_i)$; and |
39 \item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction |
29 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
40 $F: (I\times Q)\times X\to T$ also has support $S$. |
30 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
41 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
31 \end{itemize} |
42 [submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
32 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
43 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
33 of singular cells, each of which is adapted to $\cU$. |
44 (Of course we must assume that $X$ and $T$ are the appropriate |
34 \medskip |
45 sort of manifolds for this to make sense.) |
35 \hrule |
46 \end{enumerate} |
36 \medskip |
47 \end{lemma} |
37 \nn{another refugee:} |
48 |
38 |
49 |
39 We will actually prove the following more general result. |
50 |
40 Let $S$ and $T$ be an arbitrary topological spaces. |
51 |
41 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
52 \noop{ |
42 %check this; or maybe just restrict the cover} |
53 |
43 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
54 \nn{move this to later:} |
44 from $S$ to $T$. |
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45 Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
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46 \nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
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47 in any case, in our applications $S$ will always be a manifold} |
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48 |
55 |
49 \begin{lemma} \label{extension_lemma_b} |
56 \begin{lemma} \label{extension_lemma_b} |
50 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
57 Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
51 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
58 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
52 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
59 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
53 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
60 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
54 chains of smooth maps or immersions. |
61 chains of smooth maps or immersions. |
55 \end{lemma} |
62 \end{lemma} |
250 |
263 |
251 \nn{this completes proof} |
264 \nn{this completes proof} |
252 |
265 |
253 \end{proof} |
266 \end{proof} |
254 |
267 |
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268 |
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269 |
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270 |
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271 \medskip |
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272 \hrule |
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273 \medskip |
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274 \nn{the following was removed from earlier section; it should be reincorporated somehwere |
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275 in this section} |
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276 |
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277 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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278 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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279 {\it adapted to $\cU$} if there is a factorization |
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280 \eq{ |
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281 P = P_1 \times \cdots \times P_m |
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282 } |
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283 (for some $m \le k$) |
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284 and families of homeomorphisms |
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285 \eq{ |
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286 f_i : P_i \times X \to X |
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287 } |
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288 such that |
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289 \begin{itemize} |
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290 \item each $f_i$ is supported on some connected $V_i \sub X$; |
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291 \item the sets $V_i$ are mutually disjoint; |
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292 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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293 where $k_i = \dim(P_i)$; and |
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294 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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295 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
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296 \end{itemize} |
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297 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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298 of singular cells, each of which is adapted to $\cU$. |
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299 \medskip |
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300 \hrule |
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301 \medskip |
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302 |
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303 |
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304 |
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305 |
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306 |
255 \input{text/appendixes/explicit.tex} |
307 \input{text/appendixes/explicit.tex} |
256 |
308 |