author | Kevin Walker <kevin@canyon23.net> |
Tue, 18 May 2010 22:49:17 -0600 | |
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parent 271 | cb40431c8a65 |
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permissions | -rw-r--r-- |
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%!TEX root = ../../blob1.tex |
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\section{Adapting families of maps to open covers} \label{sec:localising} |
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Let $X$ and $T$ be topological spaces, with $X$ compact. |
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Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
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unity $\{r_\alpha\}$. |
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(That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
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for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
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Since $X$ is compact, we will further assume that $r_\alpha \ne 0$ (globally) |
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for only finitely |
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many $\alpha$. |
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Let |
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\[ |
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CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
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\] |
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the singular chains on the space of continuous maps from $X$ to $T$. |
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$CM_k(X, T)$ is generated by continuous maps |
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\[ |
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f: P\times X \to T , |
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\] |
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where $P$ is some convex linear polyhedron in $\r^k$. |
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Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
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$x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
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$f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
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A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
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generators which are adapted. |
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\begin{lemma} \label{basic_adaptation_lemma} |
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The $f: P\times X \to T$, as above. |
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The there exists |
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\[ |
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F: I \times P\times X \to T |
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\] |
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such that |
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\begin{enumerate} |
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\item $F(0, \cdot, \cdot) = f$ . |
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\item We can decompose $P = \cup_i D_i$ so that |
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the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
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\item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction |
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$F: (I\times Q)\times X\to T$ also has support $S$. |
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\item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
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[submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
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then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
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(Of course we must assume that $X$ and $T$ are the appropriate |
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sort of manifolds for this to make sense.) |
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\end{enumerate} |
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\end{lemma} |
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\begin{proof} |
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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 makes the conclusions of the lemma true. |
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For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$ |
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such that the various $K_\alpha$ are in general position with respect to each other. |
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If we are in one of the cases of item 4 of the lemma, also choose $K_\alpha$ |
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sufficiently fine as described below. |
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\def\jj{\tilde{L}} |
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Let $L$ be a common refinement all the $K_\alpha$'s. |
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Let $\jj$ denote the handle decomposition of $P$ corresponding to $L$. |
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Each $i$-handle $C$ of $\jj$ has an $i$-dimensional tangential coordinate and, |
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more importantly for our purposes, a $k{-}i$-dimensional normal coordinate. |
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We will typically use the same notation for $i$-cells of $L$ and the |
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corresponding $i$-handles of $\jj$. |
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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 $\jj$. |
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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 convex linear polyhedron, 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 $\jj$. |
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For handles of $\jj$ 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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\nn{*** resume revising here ***} |
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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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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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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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\nn{finally, general case...} |
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\nn{this completes proof} |
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\end{proof} |
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\noop{ |
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\nn{move this to later:} |
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\begin{lemma} \label{extension_lemma_b} |
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Let $x \in CM_k(X, T)$ 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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\nn{the following was removed from earlier section; it should be reincorporated somewhere |
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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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\input{text/appendixes/explicit.tex} |
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