author | Kevin Walker <kevin@canyon23.net> |
Sat, 22 May 2010 12:17:23 -0600 | |
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parent 274 | 8e021128cf8f |
child 276 | 7a67f45e2475 |
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 $f$ has support $S\sub X$, then |
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$F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) 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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[immersion, 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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For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ |
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and let $p(D, \alpha) = p(C(D, \alpha))$. |
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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(D, \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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Thus far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $\jj$. |
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We will now extend $u$ inductively to handles of index less than $k$. |
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Let $E$ be a $k{-}1$-handle. |
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$E$ is homeomorphic to $B^{k-1}\times [0,1]$, and meets |
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the $k$-handles at $B^{k-1}\times\{0\}$ and $B^{k-1}\times\{1\}$. |
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Let $\eta : E \to [0,1]$, $\eta(x, s) = s$ be the normal coordinate |
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of $E$. |
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Let $D_0$ and $D_1$ be the two $k$-handles of $\jj$ adjacent to $E$. |
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There is at most one index $\beta$ such that $C(D_0, \beta) \ne C(D_1, \beta)$. |
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(If there is no such index, choose $\beta$ |
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arbitrarily.) |
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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(D_0, \alpha) |
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+ r_\beta(x) (\eta(p) p(D_0, \beta) + (1-\eta(p)) p(D_1, \beta)) \right) . |
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} |
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Now for the general case. |
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Let $E$ be a $k{-}j$-handle. |
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Let $D_0,\ldots,D_a$ be the $k$-handles adjacent to $E$. |
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There is a subset of cover indices $\cN$, of cardinality $j$, |
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such that if $\alpha\notin\cN$ then |
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$p(D_u, \alpha) = p(D_v, \alpha)$ for all $0\le u,v \le a$. |
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For fixed $\beta\in\cN$ let $\{q_{\beta i}\}$ be the set of values of |
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$p(D_u, \beta)$ for $0\le u \le a$. |
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Recall the product structure $E = B^{k-j}\times B^j$. |
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Inductively, we have defined functions $\eta_{\beta i}:\bd B^j \to [0,1]$ such that |
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$\sum_i \eta_{\beta i} = 1$ for all $\beta\in \cN$. |
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Choose extensions of $\eta_{\beta i}$ to all of $B^j$. |
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Via the projection $E\to B^j$, regard $\eta_{\beta i}$ as a function on $E$. |
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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(D_0, \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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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 $u$ affords $F$ the properties claimed in the statement of the lemma. |
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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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\medskip |
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Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ |
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is a singular cell adapted to $\cU$. |
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Let $k-j$ be the index of $D$. |
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Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if |
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$r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if |
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$x\in \bigcup_{\beta\in\cN} U_\beta$. |
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Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, |
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this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$. |
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\medskip |
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Next we show that $F$ does not increase supports. |
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If $f(p,x) = f(p',x)$ for all $p,p'\in P$, |
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then |
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\[ |
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F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x) |
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\] |
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for all $(t,p)$ and $(t',p')$ in $I\times P$. |
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\medskip |
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Now for claim 4 of the lemma. |
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Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. |
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We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a |
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diffeomorphism for all $t$ and $p$. |
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It suffices to |
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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 and $X$ and $P$ are compact, |
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$\pd{f}{x}$ is non-singular and bounded away from zero. |
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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 $p(D_0,\alpha)$'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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If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes |
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through essentially unchanged. |
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Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. |
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We assume that $f$ is Lipschitz in $P$ direction as well. |
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The argument in this case is similar to the one above for diffeomorphisms, with |
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bounded partial derivatives replaced by Lipschitz constants. |
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Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for |
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$f(p, \cdot)$ for all $p$. |
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By choosing the cell decompositions $K_\alpha$ sufficiently fine, |
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we can insure that $u$ has a small Lipschitz constant in the $X$ direction. |
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This allows us to show that $F(t, p, \cdot)$ has a bi-Lipschitz constant |
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close to the universal bi-Lipschitz constant for $f$. |
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Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
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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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\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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} |
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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 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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