|
1 %!TEX root = ../blob1.tex |
|
2 |
|
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
|
4 |
|
5 Lo, the proof of Lemma (\ref{extension_lemma}): |
|
6 |
|
7 \nn{should this be an appendix instead?} |
|
8 |
|
9 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
10 later draft} |
|
11 |
|
12 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
13 about boundary later} |
|
14 |
|
15 Recall that we are given |
|
16 an open cover $\cU = \{U_\alpha\}$ and an |
|
17 $x \in CD_k(X)$ 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 $\cU$. |
|
19 |
|
20 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
21 |
|
22 As a first approximation to the argument we will eventually make, let's replace $x$ |
|
23 with a single singular cell |
|
24 \eq{ |
|
25 f: P \times X \to X . |
|
26 } |
|
27 Also, we'll ignore for now issues around $\bd P$. |
|
28 |
|
29 Our homotopy will have the form |
|
30 \eqar{ |
|
31 F: I \times P \times X &\to& X \\ |
|
32 (t, p, x) &\mapsto& f(u(t, p, x), x) |
|
33 } |
|
34 for some function |
|
35 \eq{ |
|
36 u : I \times P \times X \to P . |
|
37 } |
|
38 First we describe $u$, then we argue that it does what we want it to do. |
|
39 |
|
40 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
41 The various $K_\alpha$ should be in general position with respect to each other. |
|
42 We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
43 to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
44 merely a homotopy through maps $X\to X$. |
|
45 |
|
46 Let $L$ be the union of all the $K_\alpha$'s. |
|
47 $L$ is itself a cell decomposition of $P$. |
|
48 \nn{next two sentences not needed?} |
|
49 To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
50 where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
51 Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
52 |
|
53 Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
54 Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
55 more importantly, a $k{-}i$-dimensional normal coordinate. |
|
56 |
|
57 For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
|
58 Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
|
59 $k$-cell of $L$. |
|
60 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
61 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
62 |
|
63 For $p \in D$ we define |
|
64 \eq{ |
|
65 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
|
66 } |
|
67 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
68 makes sense.) |
|
69 |
|
70 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
71 For handles of $J$ of index less than $k$, we will define $u$ to |
|
72 interpolate between the values on $k$-handles defined above. |
|
73 |
|
74 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
|
75 of $E$. |
|
76 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
77 with a $k$-handle. |
|
78 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
79 corresponding to $E$. |
|
80 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
81 adjacent to the $k{-}1$-cell corresponding to $E$. |
|
82 For $p \in E$, define |
|
83 \eq{ |
|
84 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
|
85 + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
86 } |
|
87 |
|
88 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
89 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
90 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
91 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
92 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 |
|
94 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. |
|
96 Now define, for $p \in E$, |
|
97 \eq{ |
|
98 u(t, p, x) = (1-t)p + t \left( |
|
99 \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) |
|
101 \right) . |
|
102 } |
|
103 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
104 mentioned above. |
|
105 |
|
106 This completes the definition of $u: I \times P \times X \to P$. |
|
107 |
|
108 \medskip |
|
109 |
|
110 Next we verify that $u$ has the desired properties. |
|
111 |
|
112 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
|
113 Therefore $F$ is a homotopy from $f$ to something. |
|
114 |
|
115 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
|
116 then $F$ is a homotopy through diffeomorphisms. |
|
117 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
118 We have |
|
119 \eq{ |
|
120 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
|
121 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
122 } |
|
123 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
124 \nn{bounded away from zero, or something like that}. |
|
125 (Recall that $X$ and $P$ are compact.) |
|
126 Also, $\pd{f}{p}$ is bounded. |
|
127 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 $\pd{r_\alpha}{x}$ |
|
129 (which is bounded) |
|
130 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. |
|
132 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
133 |
|
134 \medskip |
|
135 |
|
136 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
137 is a singular cell adapted to $\cU$. |
|
138 This will complete the proof of the lemma. |
|
139 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
140 |
|
141 Let $j$ be the codimension of $D$. |
|
142 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
|
143 between handle and corresponding cell.) |
|
144 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
145 where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
146 cells of codimension greater than 0 which intersect to form $D$. |
|
147 We will show that |
|
148 if the relevant $U_\alpha$'s are disjoint, then |
|
149 $F(1, \cdot, \cdot) : D\times X \to X$ |
|
150 is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
151 If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
152 cells whose dimensions correspond to a partition of the $j_i$'s. |
|
153 We will consider some simple special cases first, then do the general case. |
|
154 |
|
155 First consider the case $j=0$ (and $m=0$). |
|
156 A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
157 is independent of $p \in P$. |
|
158 So the corresponding map $D \to \Diff(X)$ is constant. |
|
159 |
|
160 Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
161 Now Equation yyyy applies. |
|
162 We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
163 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
164 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
165 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
166 |
|
167 Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
168 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
|
169 1-dimensional. |
|
170 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
171 and a 2-cell with support $U_\beta$. |
|
172 |
|
173 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
|
174 In this case the codimension 2 cell $D$ is the intersection of two |
|
175 codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
176 We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
177 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
178 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
179 two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
180 If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
181 a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
182 \nn{need to check that this is true} |
|
183 |
|
184 \nn{finally, general case...} |
|
185 |
|
186 \nn{this completes proof} |
|
187 |
|
188 \input{text/explicit.tex} |
|
189 |