37 such that |
37 such that |
38 \begin{enumerate} |
38 \begin{enumerate} |
39 \item $F(0, \cdot, \cdot) = f$ . |
39 \item $F(0, \cdot, \cdot) = f$ . |
40 \item We can decompose $P = \cup_i D_i$ so that |
40 \item We can decompose $P = \cup_i D_i$ so that |
41 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
41 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
42 \item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction |
42 \item If $f$ has support $S\sub X$, then |
43 $F: (I\times Q)\times X\to T$ also has support $S$. |
43 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
44 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
44 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
45 [submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
45 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
46 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
46 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
47 (Of course we must assume that $X$ and $T$ are the appropriate |
47 (Of course we must assume that $X$ and $T$ are the appropriate |
48 sort of manifolds for this to make sense.) |
48 sort of manifolds for this to make sense.) |
49 \end{enumerate} |
49 \end{enumerate} |
50 \end{lemma} |
50 \end{lemma} |
51 |
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52 |
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53 |
51 |
54 \begin{proof} |
52 \begin{proof} |
55 Our homotopy will have the form |
53 Our homotopy will have the form |
56 \eqar{ |
54 \eqar{ |
57 F: I \times P \times X &\to& X \\ |
55 F: I \times P \times X &\to& X \\ |
137 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
135 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
138 |
136 |
139 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 $x$. |
137 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 $x$. |
140 Therefore $F$ is a homotopy from $f$ to something. |
138 Therefore $F$ is a homotopy from $f$ to something. |
141 |
139 |
142 \nn{*** resume revising here ***} |
140 |
143 |
141 \medskip |
144 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
142 |
145 then $F$ is a homotopy through diffeomorphisms. |
143 Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ |
146 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
144 is a singular cell adapted to $\cU$. |
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145 Let $k-j$ be the index of $D$. |
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146 Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if |
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147 $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if |
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148 $x\in \bigcup_{\beta\in\cN} U_\beta$. |
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149 Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, |
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150 this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$. |
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151 |
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152 \medskip |
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153 |
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154 Next we show that $F$ does not increase supports. |
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155 If $f(p,x) = f(p',x)$ for all $p,p'\in P$, |
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156 then |
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157 \[ |
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158 F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x) |
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159 \] |
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160 for all $(t,p)$ and $(t',p')$ in $I\times P$. |
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161 |
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162 \medskip |
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163 |
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164 Now for claim 4 of the lemma. |
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165 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. |
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166 We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a |
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167 diffeomorphism for all $t$ and $p$. |
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168 It suffices to |
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169 show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
147 We have |
170 We have |
148 \eq{ |
171 \eq{ |
149 % \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) . |
172 % \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) . |
150 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
173 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
151 } |
174 } |
152 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
175 Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, |
153 \nn{bounded away from zero, or something like that}. |
176 $\pd{f}{x}$ is non-singular and bounded away from zero. |
154 (Recall that $X$ and $P$ are compact.) |
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155 Also, $\pd{f}{p}$ is bounded. |
177 Also, $\pd{f}{p}$ is bounded. |
156 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
178 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
157 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
179 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
158 (which is bounded) |
180 (which is bounded) |
159 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
181 and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s. |
160 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
182 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
161 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
183 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
162 |
184 |
163 \medskip |
185 If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes |
164 |
186 through essentially unchanged. |
165 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
187 |
166 is a singular cell adapted to $\cU$. |
188 Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. |
167 This will complete the proof of the lemma. |
189 We assume that $f$ is Lipschitz in $P$ direction as well. |
168 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
190 The argument in this case is similar to the one above for diffeomorphisms, with |
169 |
191 bounded partial derivatives replaced by Lipschitz constants. |
170 Let $j$ be the codimension of $D$. |
192 Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for |
171 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
193 $f(p, \cdot)$ for all $p$. |
172 between handle and corresponding cell.) |
194 By choosing the cell decompositions $K_\alpha$ sufficiently fine, |
173 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
195 we can insure that $u$ has a small Lipschitz constant in the $X$ direction. |
174 where the $j_i$'s are the codimensions of the $K_\alpha$ |
196 This allows us to show that $F(t, p, \cdot)$ has a bi-Lipschitz constant |
175 cells of codimension greater than 0 which intersect to form $D$. |
197 close to the universal bi-Lipschitz constant for $f$. |
176 We will show that |
198 |
177 if the relevant $U_\alpha$'s are disjoint, then |
199 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
178 $F(1, \cdot, \cdot) : D\times X \to X$ |
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179 is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
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180 If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
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181 cells whose dimensions correspond to a partition of the $j_i$'s. |
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182 We will consider some simple special cases first, then do the general case. |
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183 |
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184 First consider the case $j=0$ (and $m=0$). |
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185 A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
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186 is independent of $p \in P$. |
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187 So the corresponding map $D \to \Diff(X)$ is constant. |
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188 |
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189 Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
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190 Now Equation yyyy applies. |
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191 We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
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192 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
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193 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
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194 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
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195 |
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196 Next case: $j=2$, $m=1$, $j_1 = 2$. |
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197 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
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198 1-dimensional. |
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199 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
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200 and a 2-cell with support $U_\beta$. |
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201 |
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202 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
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203 In this case the codimension 2 cell $D$ is the intersection of two |
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204 codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
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205 We can write $D = D' \times I \times I$, where the normal coordinates are constant |
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206 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
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207 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
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208 two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
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209 If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
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210 a 2-cell supported on $U_\beta \cup U_\gamma$. |
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211 \nn{need to check that this is true} |
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212 |
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213 \nn{finally, general case...} |
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214 |
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215 \nn{this completes proof} |
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216 |
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217 \end{proof} |
200 \end{proof} |
218 |
201 |
219 |
202 |
220 |
203 |
221 |
204 |