1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
2 |
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3 \section{Families of Diffeomorphisms} \label{sec:localising} |
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
4 |
4 |
5 Lo, the proof of Lemma (\ref{extension_lemma}): |
5 In this appendix we provide the proof of |
6 |
6 |
7 \nn{should this be an appendix instead?} |
7 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
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8 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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9 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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10 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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11 \end{lem*} |
8 |
12 |
9 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
13 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
10 later draft} |
14 later draft} |
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15 |
12 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
16 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
13 about boundary later} |
17 about boundary later} |
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18 |
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19 \begin{proof} |
14 |
20 |
15 Recall that we are given |
21 Recall that we are given |
16 an open cover $\cU = \{U_\alpha\}$ and an |
22 an open cover $\cU = \{U_\alpha\}$ and an |
17 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
23 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
18 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
24 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
92 a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
98 a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
93 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
99 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
94 the $k{-}j$-cell corresponding to $E$. |
100 the $k{-}j$-cell corresponding to $E$. |
95 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
101 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
96 Now define, for $p \in E$, |
102 Now define, for $p \in E$, |
97 \eq{ |
103 \begin{equation} |
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104 \label{eq:u} |
98 u(t, p, x) = (1-t)p + t \left( |
105 u(t, p, x) = (1-t)p + t \left( |
99 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
106 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
100 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
107 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
101 \right) . |
108 \right) . |
102 } |
109 \end{equation} |
103 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
110 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
104 mentioned above. |
111 mentioned above. |
105 |
112 |
106 This completes the definition of $u: I \times P \times X \to P$. |
113 This completes the definition of $u: I \times P \times X \to P$. |
107 |
114 |
123 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
130 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
124 \nn{bounded away from zero, or something like that}. |
131 \nn{bounded away from zero, or something like that}. |
125 (Recall that $X$ and $P$ are compact.) |
132 (Recall that $X$ and $P$ are compact.) |
126 Also, $\pd{f}{p}$ is bounded. |
133 Also, $\pd{f}{p}$ is bounded. |
127 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
134 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
128 It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
135 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
129 (which is bounded) |
136 (which is bounded) |
130 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
137 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
131 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
138 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
132 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
139 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
133 |
140 |