# HG changeset patch # User Kevin Walker # Date 1274244557 21600 # Node ID a7a23eeb5d650fd5aacfa4f37abe6b93b1b1b63a # Parent cb40431c8a65ba785abffc0a978b4129c6d12057 more famodiff.tex diff -r cb40431c8a65 -r a7a23eeb5d65 text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Sun May 16 17:15:00 2010 -0700 +++ b/text/appendixes/famodiff.tex Tue May 18 22:49:17 2010 -0600 @@ -3,11 +3,14 @@ \section{Adapting families of maps to open covers} \label{sec:localising} -Let $X$ and $T$ be topological spaces. +Let $X$ and $T$ be topological spaces, with $X$ compact. Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of unity $\{r_\alpha\}$. (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) +Since $X$ is compact, we will further assume that $r_\alpha \ne 0$ (globally) +for only finitely +many $\alpha$. Let \[ @@ -18,7 +21,7 @@ \[ f: P\times X \to T , \] -where $P$ is some linear polyhedron in $\r^k$. +where $P$ is some convex linear polyhedron in $\r^k$. Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. @@ -48,61 +51,7 @@ - -\noop{ - -\nn{move this to later:} - -\begin{lemma} \label{extension_lemma_b} -Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. -Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. -Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. -If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with -chains of smooth maps or immersions. -\end{lemma} - -\medskip -\hrule -\medskip - - -In this appendix we provide the proof of -\nn{should change this to the more general \ref{extension_lemma_b}} - -\begin{lem*}[Restatement of Lemma \ref{extension_lemma}] -Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. -Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. -Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. -\end{lem*} - -\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} - -} - - -\nn{**** resume revising here ****} - - \begin{proof} - -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 \\ @@ -112,42 +61,40 @@ \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. + +First we describe $u$, then we argue that it makes the conclusions of the lemma true. -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 $X\to X$. +For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$ +such that the various $K_\alpha$ are in general position with respect to each other. +If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$ +sufficiently fine as described below. -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 $\sum c_\alpha \le k$. +\def\jj{\tilde{L}} +Let $L$ be a common refinement all the $K_\alpha$'s. +Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$. +Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and, +more importantly for our purposes, a $k{-}i$-dimensional normal coordinate. +We will typically use the same notation for $i$-cells of $L$ and the +corresponding $i$-handles of $\jj$. -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 $L$. +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 $\jj$. 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 $P$. +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 $P$ +(Recall that $P$ is a convex linear polyhedron, 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 +So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $\jj$. +For handles of $\jj$ of index less than $k$, we will define $u$ to interpolate between the values on $k$-handles defined above. +\nn{*** resume revising here ***} + 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 $E$ @@ -268,10 +215,47 @@ +\noop{ + +\nn{move this to later:} + +\begin{lemma} \label{extension_lemma_b} +Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. +Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. +Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. +If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with +chains of smooth maps or immersions. +\end{lemma} + \medskip \hrule \medskip -\nn{the following was removed from earlier section; it should be reincorporated somehwere + + +In this appendix we provide the proof of +\nn{should change this to the more general \ref{extension_lemma_b}} + +\begin{lem*}[Restatement of Lemma \ref{extension_lemma}] +Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. +Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. +Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. +\end{lem*} + +\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} + +} + + + + +\medskip +\hrule +\medskip +\nn{the following was removed from earlier section; it should be reincorporated somewhere in this section} Let $\cU = \{U_\alpha\}$ be an open cover of $X$.