author | Kevin Walker <kevin@canyon23.net> |
Mon, 26 Apr 2010 10:43:42 -0700 | |
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parent 245 | 7537032ad5a0 |
child 271 | cb40431c8a65 |
permissions | -rw-r--r-- |
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%!TEX root = ../../blob1.tex |
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\section{Families of Diffeomorphisms} \label{sec:localising} |
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\medskip |
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\hrule |
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\medskip |
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\nn{the following was removed from earlier section; it should be reincorporated somehwere |
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in this section} |
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Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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{\it adapted to $\cU$} if there is a factorization |
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\eq{ |
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P = P_1 \times \cdots \times P_m |
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} |
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(for some $m \le k$) |
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and families of homeomorphisms |
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\eq{ |
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f_i : P_i \times X \to X |
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} |
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such that |
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\begin{itemize} |
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\item each $f_i$ is supported on some connected $V_i \sub X$; |
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\item the sets $V_i$ are mutually disjoint; |
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\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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where $k_i = \dim(P_i)$; and |
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\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
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\end{itemize} |
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A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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of singular cells, each of which is adapted to $\cU$. |
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\medskip |
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\hrule |
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\medskip |
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\nn{another refugee:} |
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We will actually prove the following more general result. |
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Let $S$ and $T$ be an arbitrary topological spaces. |
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%\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
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%check this; or maybe just restrict the cover} |
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Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
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from $S$ to $T$. |
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Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
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\nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
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in any case, in our applications $S$ will always be a manifold} |
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\begin{lemma} \label{extension_lemma_b} |
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Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
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Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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chains of smooth maps or immersions. |
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\end{lemma} |
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\medskip |
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\hrule |
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\medskip |
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In this appendix we provide the proof of |
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\nn{should change this to the more general \ref{extension_lemma_b}} |
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\begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
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Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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\end{lem*} |
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\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
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later draft} |
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\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
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about boundary later} |
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\begin{proof} |
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Recall that we are given |
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an open cover $\cU = \{U_\alpha\}$ and an |
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$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
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We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
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As a first approximation to the argument we will eventually make, let's replace $x$ |
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with a single singular cell |
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\eq{ |
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f: P \times X \to X . |
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} |
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Also, we'll ignore for now issues around $\bd P$. |
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Our homotopy will have the form |
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\eqar{ |
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F: I \times P \times X &\to& X \\ |
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(t, p, x) &\mapsto& f(u(t, p, x), x) |
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} |
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for some function |
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\eq{ |
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u : I \times P \times X \to P . |
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} |
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First we describe $u$, then we argue that it does what we want it to do. |
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For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
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The various $K_\alpha$ should be in general position with respect to each other. |
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We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
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to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
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merely a homotopy through maps $X\to X$. |
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Let $L$ be the union of all the $K_\alpha$'s. |
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$L$ is itself a cell decomposition of $P$. |
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\nn{next two sentences not needed?} |
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To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
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where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
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Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
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Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
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Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
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more importantly, a $k{-}i$-dimensional normal coordinate. |
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For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
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$k$-cell of $L$. |
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To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
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which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
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For $p \in D$ we define |
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\eq{ |
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u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
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} |
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(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
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makes sense.) |
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So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
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For handles of $J$ of index less than $k$, we will define $u$ to |
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interpolate between the values on $k$-handles defined above. |
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If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
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of $E$. |
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In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
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with a $k$-handle. |
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Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
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corresponding to $E$. |
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Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
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adjacent to the $k{-}1$-cell corresponding to $E$. |
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For $p \in E$, define |
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\eq{ |
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u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
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+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
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} |
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In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
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$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
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The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
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If we triangulate $R$ (without introducing new vertices), we can linearly extend |
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a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
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Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
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the $k{-}j$-cell corresponding to $E$. |
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For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
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Now define, for $p \in E$, |
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\begin{equation} |
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\label{eq:u} |
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u(t, p, x) = (1-t)p + t \left( |
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\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
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+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
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\right) . |
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\end{equation} |
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Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
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mentioned above. |
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This completes the definition of $u: I \times P \times X \to P$. |
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\medskip |
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Next we verify that $u$ has the desired properties. |
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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 $x$. |
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Therefore $F$ is a homotopy from $f$ to something. |
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Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
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then $F$ is a homotopy through diffeomorphisms. |
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We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
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We have |
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\eq{ |
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% \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) . |
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\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
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} |
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Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
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\nn{bounded away from zero, or something like that}. |
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(Recall that $X$ and $P$ are compact.) |
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Also, $\pd{f}{p}$ is bounded. |
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So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
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It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
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(which is bounded) |
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and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
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These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
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This completes the proof that $F$ is a homotopy through diffeomorphisms. |
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\medskip |
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Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
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is a singular cell adapted to $\cU$. |
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This will complete the proof of the lemma. |
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\nn{except for boundary issues and the `$P$ is a cell' assumption} |
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Let $j$ be the codimension of $D$. |
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(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
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between handle and corresponding cell.) |
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Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
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where the $j_i$'s are the codimensions of the $K_\alpha$ |
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cells of codimension greater than 0 which intersect to form $D$. |
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We will show that |
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if the relevant $U_\alpha$'s are disjoint, then |
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$F(1, \cdot, \cdot) : D\times X \to X$ |
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is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
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If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
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cells whose dimensions correspond to a partition of the $j_i$'s. |
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We will consider some simple special cases first, then do the general case. |
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First consider the case $j=0$ (and $m=0$). |
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A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
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is independent of $p \in P$. |
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So the corresponding map $D \to \Diff(X)$ is constant. |
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Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
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Now Equation yyyy applies. |
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We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
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It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
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of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
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The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
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232 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
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This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
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1-dimensional. |
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We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
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and a 2-cell with support $U_\beta$. |
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238 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
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In this case the codimension 2 cell $D$ is the intersection of two |
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codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
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We can write $D = D' \times I \times I$, where the normal coordinates are constant |
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on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
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If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
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two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
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If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
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a 2-cell supported on $U_\beta \cup U_\gamma$. |
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\nn{need to check that this is true} |
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248 |
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249 |
\nn{finally, general case...} |
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250 |
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251 |
\nn{this completes proof} |
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194 | 253 |
\end{proof} |
98 | 254 |
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194 | 255 |
\input{text/appendixes/explicit.tex} |
256 |