1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
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3 \section{Families of Diffeomorphisms} \label{sec:localising} |
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
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4 |
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5 |
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6 \medskip |
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7 \hrule |
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8 \medskip |
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9 \nn{the following was removed from earlier section; it should be reincorporated somehwere |
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10 in this section} |
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11 |
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12 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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13 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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14 {\it adapted to $\cU$} if there is a factorization |
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15 \eq{ |
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16 P = P_1 \times \cdots \times P_m |
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17 } |
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18 (for some $m \le k$) |
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19 and families of homeomorphisms |
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20 \eq{ |
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21 f_i : P_i \times X \to X |
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22 } |
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23 such that |
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24 \begin{itemize} |
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25 \item each $f_i$ is supported on some connected $V_i \sub X$; |
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26 \item the sets $V_i$ are mutually disjoint; |
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27 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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28 where $k_i = \dim(P_i)$; and |
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29 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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30 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
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31 \end{itemize} |
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32 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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33 of singular cells, each of which is adapted to $\cU$. |
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34 \medskip |
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35 \hrule |
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36 \medskip |
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37 \nn{another refugee:} |
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38 |
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39 We will actually prove the following more general result. |
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40 Let $S$ and $T$ be an arbitrary topological spaces. |
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41 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
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42 %check this; or maybe just restrict the cover} |
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43 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
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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 |
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49 \begin{lemma} \label{extension_lemma_b} |
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50 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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51 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
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52 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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53 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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54 chains of smooth maps or immersions. |
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55 \end{lemma} |
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56 |
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57 \medskip |
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58 \hrule |
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59 \medskip |
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60 |
4 |
61 |
5 In this appendix we provide the proof of |
62 In this appendix we provide the proof of |
6 \nn{should change this to the more general \ref{extension_lemma_b}} |
63 \nn{should change this to the more general \ref{extension_lemma_b}} |
7 |
64 |
8 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
65 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |