blob: comparison text/appendixes/famodiff.tex

equal deleted inserted replaced

-:ba4ddbc41c15
+:79f7b1bd7b1a
 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)
+Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally)
-for only finitely
+for all but finitely many $\alpha$.
-many $\alpha$.
 Let
 \[
 	CM_*(X, T) \deq C_*(\Maps(X\to T)) ,
 \]
 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s.
 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of
 generators which are adapted.
 \begin{lemma} \label{basic_adaptation_lemma}
-The $f: P\times X \to T$, as above.
+Let $f: P\times X \to T$, as above.
-The there exists
+Then there exists
 \[
 	F: I \times P\times X \to T
 \]
 such that
 \begin{enumerate}
 \item $F(0, \cdot, \cdot) = f$ .
 \item We can decompose $P = \cup_i D_i$ so that
 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$.
 \item If $f$ has support $S\sub X$, then
 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$.
+Furthermore, if $Q\sub\bd P$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$
+has support $S'$, then
+$F: (I\times Q)\times X\to T$ also has support $S'$.
 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a
 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism]
 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$.
 (Of course we must assume that $X$ and $T$ are the appropriate
 sort of manifolds for this to make sense.)
 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.
 \def\jj{\tilde{L}}
-Let $L$ be a common refinement all the $K_\alpha$'s.
+Let $L$ be a common refinement of 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 $\jj$.
 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$.
+If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$.
+(It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$.
+This puts some mild constraints on the choice of $K_\alpha$.)
 Let $D$ be a $k$-handle of $\jj$.
 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$
 and let $p(D, \alpha) = p(C(D, \alpha))$.
 For $p \in D$ we define
 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
 \right) .
 \end{equation}
 This completes the definition of $u: I \times P \times X \to P$.
+Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$.
 \medskip
 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma.
 then
 \[
 	F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x)
 \]
 for all $(t,p)$ and $(t',p')$ in $I\times P$.
+Similarly, if $f(q,x) = f(q',x)$ for all $q,q'\in Q\sub \bd P$,
+then
+\[
+	F(t, q, x) = f(u(t,q,x),x) = f(u(t',q',x),x) = F(t',q',x)
+\]
+for all $(t,q)$ and $(t',q')$ in $I\times Q$.
+(Recall that we arranged above that $u(I\times Q\times X) \sub Q$.)
 \medskip
 Now for claim 4 of the lemma.
 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms.
 from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$
 of $X$.
 Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$.
 \end{lemma}
 \begin{proof}
-\nn{my current idea is too messy, so I'm going to wait and hopefully think
+If suffices to show that given a generator $f:P\times X\to T$ of $CM_k(X\to T)$ with
-of a cleaner proof}
+$\bd f \in G_{k-1}$ there exists $h\in CM_{k+1}(X\to T)$ with $\bd h = f + g$ and $g \in G_k$.
-\noop{
+This is exactly what Lemma \ref{basic_adaptation_lemma}
-If suffices to show that
+gives us.
-...
+More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$.
-Lemma \ref{basic_adaptation_lemma}
+Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}.
-...
+Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$.
-}
+Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$,
+while part 3 of Lemma \ref{basic_adaptation_lemma} says that the restrictions to $I\times Q_i$ are in $G_k$.
 \end{proof}
 \medskip
 \nn{need to clean up references from the main text to the lemmas of this section}
+%%%%%% Lo, \noop{...}
 \noop{
 \medskip
 \nn{do we want to keep the following?}

changeset 278	79f7b1bd7b1a
parent 277	ba4ddbc41c15
child 281	630ceb40a07b