diff -r 7a880cdaac70 -r 395bd663e20d text/famodiff.tex --- a/text/famodiff.tex Fri Oct 23 04:12:41 2009 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,189 +0,0 @@ -%!TEX root = ../blob1.tex - -\section{Families of Diffeomorphisms} \label{sec:localising} - -Lo, the proof of Lemma (\ref{extension_lemma}): - -\nn{should this be an appendix instead?} - -\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in -later draft} - -\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry -about boundary later} - -Recall that we are given -an open cover $\cU = \{U_\alpha\}$ and an -$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. -We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. - -Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. - -As a first approximation to the argument we will eventually make, let's replace $x$ -with a single singular cell -\eq{ - f: P \times X \to X . -} -Also, we'll ignore for now issues around $\bd P$. - -Our homotopy will have the form -\eqar{ - F: I \times P \times X &\to& X \\ - (t, p, x) &\mapsto& f(u(t, p, x), x) -} -for some function -\eq{ - u : I \times P \times X \to P . -} -First we describe $u$, then we argue that it does what we want it to do. - -For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. -The various $K_\alpha$ should be in general position with respect to each other. -We will see below that the $K_\alpha$'s need to be sufficiently fine in order -to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not -merely a homotopy through maps $X\to X$. - -Let $L$ be the union of all the $K_\alpha$'s. -$L$ is itself a cell decomposition of $P$. -\nn{next two sentences not needed?} -To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, -where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. -Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. - -Let $J$ denote the handle decomposition of $P$ corresponding to $L$. -Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, -more importantly, a $k{-}i$-dimensional normal coordinate. - -For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. -Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding -$k$-cell of $L$. -To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s -which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. - -For $p \in D$ we define -\eq{ - u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . -} -(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ -makes sense.) - -So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. -For handles of $J$ of index less than $k$, we will define $u$ to -interpolate between the values on $k$-handles defined above. - -If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate -of $E$. -In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ -with a $k$-handle. -Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell -corresponding to $E$. -Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ -adjacent to the $k{-}1$-cell corresponding to $E$. -For $p \in E$, define -\eq{ - u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} - + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . -} - -In general, for $E$ a $k{-}j$-handle, there is a normal coordinate -$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. -The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. -If we triangulate $R$ (without introducing new vertices), we can linearly extend -a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. -Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets -the $k{-}j$-cell corresponding to $E$. -For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. -Now define, for $p \in E$, -\eq{ - u(t, p, x) = (1-t)p + t \left( - \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} - + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) - \right) . -} -Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension -mentioned above. - -This completes the definition of $u: I \times P \times X \to P$. - -\medskip - -Next we verify that $u$ has the desired properties. - -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. - -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 $(t, p, x)$. -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.) -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 xxxx 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. -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. - -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 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} - -\input{text/explicit.tex} -