diff -r e924dd389d6e -r ec3af8dfcb3c text/famodiff.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/famodiff.tex Tue Jul 21 16:21:20 2009 +0000 @@ -0,0 +1,189 @@ +%!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} +