# HG changeset patch # User Kevin Walker # Date 1274552243 21600 # Node ID 81d7c550b3da3fcbcc8a2d5d4d0253fd34a368f4 # Parent 8e021128cf8fea3d679d57e5303f0b37d121c3bf finshed proof of main lemma in famodiff appendix diff -r 8e021128cf8f -r 81d7c550b3da text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Fri May 21 15:27:45 2010 -0600 +++ b/text/appendixes/famodiff.tex Sat May 22 12:17:23 2010 -0600 @@ -39,18 +39,16 @@ \item $F(0, \cdot, \cdot) = f$ . \item We can decompose $P = \cup_i D_i$ so that the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. -\item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction -$F: (I\times Q)\times X\to T$ also has support $S$. +\item If $f$ has support $S\sub X$, then +$F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a -[submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] +[immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. (Of course we must assume that $X$ and $T$ are the appropriate sort of manifolds for this to make sense.) \end{enumerate} \end{lemma} - - \begin{proof} Our homotopy will have the form \eqar{ @@ -104,7 +102,7 @@ For $p \in E$, define \eq{ u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p(D_0, \alpha) - + r_\beta(x) (\eta(p) p(D_0, p) + (1-\eta(p)) p(D_1, p)) \right) . + + r_\beta(x) (\eta(p) p(D_0, \beta) + (1-\eta(p)) p(D_1, \beta)) \right) . } @@ -125,7 +123,7 @@ \begin{equation} \label{eq:u} u(t, p, x) = (1-t)p + t \left( - \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} + \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha) + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) \right) . \end{equation} @@ -139,81 +137,66 @@ 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 $x$. Therefore $F$ is a homotopy from $f$ to something. -\nn{*** resume revising here ***} + +\medskip + +Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ +is a singular cell adapted to $\cU$. +Let $k-j$ be the index of $D$. +Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if +$r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if +$x\in \bigcup_{\beta\in\cN} U_\beta$. +Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, +this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$. + +\medskip -Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, -then $F$ is a homotopy through diffeomorphisms. -We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. +Next we show that $F$ does not increase supports. +If $f(p,x) = f(p',x)$ for all $p,p'\in P$, +then +\[ + F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x) +\] +for all $(t,p)$ and $(t',p')$ in $I\times P$. + +\medskip + +Now for claim 4 of the lemma. +Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. +We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a +diffeomorphism for all $t$ and $p$. +It suffices to +show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. We have \eq{ % \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) . \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . } -Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and -\nn{bounded away from zero, or something like that}. -(Recall that $X$ and $P$ are compact.) +Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, +$\pd{f}{x}$ is non-singular and bounded away from zero. Also, $\pd{f}{p}$ is bounded. So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ (which is bounded) -and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. +and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s. These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. This completes the proof that $F$ is a homotopy through diffeomorphisms. -\medskip - -Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ -is a singular cell adapted to $\cU$. -This will complete the proof of the lemma. -\nn{except for boundary issues and the `$P$ is a cell' assumption} - -Let $j$ be the codimension of $D$. -(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction -between handle and corresponding cell.) -Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, -where the $j_i$'s are the codimensions of the $K_\alpha$ -cells of codimension greater than 0 which intersect to form $D$. -We will show that -if the relevant $U_\alpha$'s are disjoint, then -$F(1, \cdot, \cdot) : D\times X \to X$ -is a product of singular cells of dimensions $j_1, \ldots, j_m$. -If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular -cells whose dimensions correspond to a partition of the $j_i$'s. -We will consider some simple special cases first, then do the general case. - -First consider the case $j=0$ (and $m=0$). -A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, -is independent of $p \in P$. -So the corresponding map $D \to \Diff(X)$ is constant. +If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes +through essentially unchanged. -Next consider the case $j = 1$ (and $m=1$, $j_1=1$). -Now Equation yyyy applies. -We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. -It follows that the singular cell $D \to \Diff(X)$ can be written as a product -of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. -The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. - -Next case: $j=2$, $m=1$, $j_1 = 2$. -This is similar to the previous case, except that the normal bundle is 2-dimensional instead of -1-dimensional. -We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell -and a 2-cell with support $U_\beta$. +Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. +We assume that $f$ is Lipschitz in $P$ direction as well. +The argument in this case is similar to the one above for diffeomorphisms, with +bounded partial derivatives replaced by Lipschitz constants. +Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for +$f(p, \cdot)$ for all $p$. +By choosing the cell decompositions $K_\alpha$ sufficiently fine, +we can insure that $u$ has a small Lipschitz constant in the $X$ direction. +This allows us to show that $F(t, p, \cdot)$ has a bi-Lipschitz constant +close to the universal bi-Lipschitz constant for $f$. -Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. -In this case the codimension 2 cell $D$ is the intersection of two -codimension 1 cells, from $K_\beta$ and $K_\gamma$. -We can write $D = D' \times I \times I$, where the normal coordinates are constant -on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. -If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and -two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. -If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and -a 2-cell supported on $U_\beta \cup U_\gamma$. -\nn{need to check that this is true} - -\nn{finally, general case...} - -\nn{this completes proof} - +Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. \end{proof}