# HG changeset patch # User Kevin Walker # Date 1313003519 21600 # Node ID 933a93ef7df1fa80250b99b3bf8c719a715500b4 # Parent 0adb2c01388032c680e059bd63f9bc5d510c5daf intermediate commit -- not done yet diff -r 0adb2c013880 -r 933a93ef7df1 text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Wed Aug 10 11:08:14 2011 -0600 +++ b/text/appendixes/famodiff.tex Wed Aug 10 13:11:59 2011 -0600 @@ -251,6 +251,10 @@ The sense of ``small" we mean will be explained below. It depends only on $\cU$ and the choice of $U_\alpha^i$'s. +Our goal is to homotope $P$, rel boundary, so that it is adapted to $\cU$. +By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), +it suffices to find $f':P\to \Homeo(X)$ such that $f' = f$ on $\bd P$ and $f'$ is adapted to $\cU$. + We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$. So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$ @@ -261,51 +265,67 @@ W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i . \] -We will construct a sequence of maps $f_i : P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: +By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), + +We will construct a sequence of maps $f_i : \bd P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: \begin{itemize} -\item[(A)] $f_0 = f$; +\item[(A)] $f_0 = f|_{\bd P}$; \item[(B)] $f_i = g$ on $W_i^i$; \item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and -\item[(D)] there is a homotopy $F_i : P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the +\item[(D)] there is a homotopy $F_i : \bd P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$. \nn{check this when done writing} \end{itemize} Once we have the $F_i$'s as in (D), we can finish the argument as follows. -Assemble the $F_i$'s into a map $F: P\times [0,N] \to \Homeo(X)$. -View $F$ as a homotopy rel boundary from $F$ restricted to $P\times\{0\}$ (which is just $f$ by (A)) -to $F$ restricted to $\bd P \times [0,N] \cup P\times\{N\}$. -$F$ restricted to $\bd P \times [0,N]$ is adapted to $\cU$ by (D). -$F$ restricted to $P\times\{N\}$ is constant on $W_N^N = X$ by (B), and therefore is, {\it a fortiori}, also adapted -to $\cU$. +Assemble the $F_i$'s into a map $F: \bd P\times [0,N] \to \Homeo(X)$. +$F$ is adapted to $\cU$ by (D). +$F$ restricted to $\bd P\times\{N\}$ is constant on $W_N^N = X$ by (B). +We can therefore view $F$ as a map $f'$ from $\Cone(\bd P) \cong P$ to $\Homeo(X)$ +which is adapted to $\cU$. + +The homotopies $F_i$ will be composed of three types of pieces, $A_\beta$, $B_\beta$ and $C$, % NOT C_\beta +as illustrated in Figure \nn{xxxx}. +($A_\beta$, $B_\beta$ and $C$ also depend on $i$, but we are suppressing that from the notation.) +The homotopy $A_\beta : Q_\beta \times I \to \Homeo(X)$ will arrange that $f_i$ agrees with $g$ +on $U_i^i \setmin V_\beta^{N-i+1}$. +The homotopy $B_\beta : Q_\beta \times I \to \Homeo(X)$ will extend the agreement with $g$ to all of $U_i^i$. +The homotopies $C$ match things up between $\bd Q_\beta \times I$ and $\bd Q_{\beta'} \times I$ when +$Q_\beta$ and $Q_{\beta'}$ are adjacent. + +Assume inductively that we have defined $f_{i-1}$. + +Now we define $A_\beta$. +Choose $q_0\in Q_\beta$. +Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$ such that +\begin{itemize} +\item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and +\item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$. +\end{itemize} +Define $A_\beta$ by +\[ + A_\beta(q, t) = h(t) \circ f_{i-1}(q) . +\] +It follows that +\begin{itemize} +\item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$; +\item[(H)] $A_\beta(q, 1) = g$ on $(U_i^i \cup W_{i-1}^{i-\frac12})\setmin V_\beta^{N-i+1}$; and +\item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$. +\end{itemize} + + + \nn{resume revising here} -Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us -to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: -\begin{itemize} -\item $h(p, 0) = f(p)$ for all $p\in P$. -\item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$, -for all $p\in P$. -\item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on -$U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$ -(and hence supported on $U_i$). -\end{itemize} + +\nn{scraps:} + +Theorem 5.1 of \cite{MR0283802}, + To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, and the subdivision mentioned above is chosen fine enough to insure this. -By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$ -to the family -\[ - h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) . -\] -We claim that the latter family of homeomorphisms is adapted to $\cU$. -By the second bullet above, $h(P\times \{N\})$ is the constant family $g$, and is therefore supported on the empty set. -Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is -adapted to $\cU$, which means that $\bd P = \cup_j Q_j$ with $f(Q_j)$ supported on the union of $k-1$ -of the $U_\alpha$'s for each $j$. -It follows (using the third bullet above) that $h(Q_j \times [i-1,i])$ is supported on the union of $k$ -of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$. \end{proof}