# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1308168919 21600
# Node ID 24f14faacab4282081d5088011400d914fd6446d
# Parent  adfffac7c138efffd1cc48f65ca54bad6db72c8a
finished topological case of Appendix B

diff -r adfffac7c138 -r 24f14faacab4 blob to-do
--- a/blob to-do	Tue Jun 14 19:28:48 2011 -0600
+++ b/blob to-do	Wed Jun 15 14:15:19 2011 -0600
@@ -1,5 +1,3 @@
-
-* extend localization lemma to (topological) homeos
 
 * lemma [inject  6.3.5?] assumes more splittablity than the axioms imply (?)
 
@@ -21,6 +19,8 @@
 
 * make sure we are clear that boundary = germ
 
+* go through text and remove any disclaimers about continuous (as oppsed to PL) homeos
+
 * review colors in figures
 
 * maybe say something in colimit section about restriction to submanifolds and submanifolds of boundary (we use this in n-cat axioms)
diff -r adfffac7c138 -r 24f14faacab4 blob_changes_v3
--- a/blob_changes_v3	Tue Jun 14 19:28:48 2011 -0600
+++ b/blob_changes_v3	Wed Jun 15 14:15:19 2011 -0600
@@ -25,7 +25,7 @@
 - strengthened n-cat isotopy invariance axiom to allow for homeomorphisms which act trivially elements on the restriction of an n-morphism to the boundary of the ball
 - more details on axioms for enriched n-cats
 - added details to the construction of traditional 1-categories from disklike 1-categories (Appendix C.1)
-- 
+- extended the lemmas of Appendix B (about adapting families of homeomorphisms to open covers) to the topological category
 
 
 
diff -r adfffac7c138 -r 24f14faacab4 text/appendixes/famodiff.tex
--- a/text/appendixes/famodiff.tex	Tue Jun 14 19:28:48 2011 -0600
+++ b/text/appendixes/famodiff.tex	Wed Jun 15 14:15:19 2011 -0600
@@ -235,17 +235,47 @@
 
 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$.
 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$
-such that $g\circ f(P)$ is a small neighborhood of the 
+such that $g^{-1}\circ f(P)$ is a small neighborhood of the 
 identity in $\Homeo(X)$.
 The sense of ``small" we mean will be explained below.
 It depends only on $\cU$ and some auxiliary covers.
 
-We assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.
+
+Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite.
+Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$.
 
+We will need some wiggle room, so for each $\alpha$ choose open sets
+\[
+	U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset \cdots \supset U_\alpha^N
+\]
+so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that
+the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$.
 
+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 $U_i^{i-1}$
+(and hence supported on $U_i$).
+\end{itemize}
+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.
 
-\nn{...}
-
+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}