# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1276431991 -7200
# Node ID 6224e50c9311afe7aa589bb3959aae93e09872a1
# Parent  8589275ac65b60a99689be0838d0e2f45c412854
metric independence for homeo action (proof done now)

diff -r 8589275ac65b -r 6224e50c9311 text/evmap.tex
--- a/text/evmap.tex	Thu Jun 10 22:00:06 2010 +0200
+++ b/text/evmap.tex	Sun Jun 13 14:26:31 2010 +0200
@@ -602,24 +602,19 @@
 
 Finally we show that the action maps defined above are independent of
 the choice of metric (up to iterated homotopy).
-
-\nn{...}
-
-
-\medskip\hrule\medskip\hrule\medskip
-
-\nn{outline of what remains to be done:}
-
-\begin{itemize}
-\item Independence of metric, $\ep_i$, $\delta_i$:
-For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes
-and $\hat{N}_{i,l}$ the alternate neighborhoods.
-Main idea is that for all $i$ there exists sufficiently large $k$ such that
-$\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed.
-\end{itemize}
-
-\nn{to be continued....}
-
+The arguments are very similar to ones given above, so we only sketch them.
+Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding
+actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$.
+We must show that $e$ and $e'$ are homotopic.
+As outlined in the discussion preceding this proof,
+this follows from the facts that both $e$ and $e'$ are compatible
+with gluing and that $\bc_*(B^n)$ is contractible.
+As above, we define a subcomplex $F_*\sub  CH_*(X, X) \ot \bc_*(X)$ generated
+by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls.
+Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$.
+We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$.
+Similar arguments show that this homotopy from $e$ to $e'$ is well-defined
+up to second order homotopy, and so on.
 \end{proof}
 
 
diff -r 8589275ac65b -r 6224e50c9311 text/ncat.tex
--- a/text/ncat.tex	Thu Jun 10 22:00:06 2010 +0200
+++ b/text/ncat.tex	Sun Jun 13 14:26:31 2010 +0200
@@ -82,7 +82,7 @@
 The 0-sphere is unusual among spheres in that it is disconnected.
 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
 (Actually, this is only true in the oriented case, with 1-morphisms parameterized
-by oriented 1-balls.)
+by {\it oriented} 1-balls.)
 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense.
 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. 
 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary.