# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1244223714 0
# Node ID 4888269574d9356ecee90808f6dbfb4d8cfda272
# Parent  ed2594ff5870e3448e085a78c26708b37c2eecc7
...

diff -r ed2594ff5870 -r 4888269574d9 text/article_preamble.tex
--- a/text/article_preamble.tex	Fri Jun 05 16:17:31 2009 +0000
+++ b/text/article_preamble.tex	Fri Jun 05 17:41:54 2009 +0000
@@ -1,6 +1,8 @@
 %auto-ignore
 %this ensures the arxiv doesn't try to start TeXing here.
 
+%!TEX root = ../blob1.tex
+
 \input{\pathtotrunk preamble.tex}
 
 \usepackage{breakurl}
diff -r ed2594ff5870 -r 4888269574d9 text/evmap.tex
--- a/text/evmap.tex	Fri Jun 05 16:17:31 2009 +0000
+++ b/text/evmap.tex	Fri Jun 05 17:41:54 2009 +0000
@@ -100,6 +100,7 @@
 
 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.
 
+%Suppose for the moment that evaluation maps with the advertised properties exist.
 Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
 Suppose that there exists $V \sub X$ such that
 \begin{enumerate}
@@ -112,12 +113,41 @@
 	p = \gl(q, r),
 \]
 where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$.
+We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$.
 According to the commutative diagram of the proposition, we must have
 \[
-	e_X(p) = e_X(\gl(q, r)) = gl(e_{VV'}(q), e_{WW'}(r)) .
+	e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = 
+				gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .
+\]
+Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have
+\[
+	e_{WW'}(r\otimes b_W) = r(b_W),
 \]
-\nn{need to add blob parts to above}
-Since $r$ is a plain, 0-parameter family of diffeomorphisms, 
+where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in
+this case a 0-blob diagram).
+Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ 
+(by \ref{disjunion} and \ref{bcontract}).
+Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
+there is, up to homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
+such that 
+\[
+	\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) .
+\]
+
+Thus the conditions of the proposition determine (up to homotopy) the evaluation
+map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint
+union of balls.
+On the other hand, Lemma \ref{extension_lemma} allows us to homotope 
+\nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property.
+\nn{should give a name to this property}
+This (roughly) establishes the uniqueness part of the proposition.
+To show existence, we must show that the various choices involved in constructing
+evaluation maps in this way affect the final answer only by a homotopy.
+
+\nn{now for a more detailed outline of the proof...}
+
+
+
 \medskip
 
 \nn{to be continued....}
diff -r ed2594ff5870 -r 4888269574d9 text/explicit.tex
--- a/text/explicit.tex	Fri Jun 05 16:17:31 2009 +0000
+++ b/text/explicit.tex	Fri Jun 05 17:41:54 2009 +0000
@@ -1,6 +1,5 @@
 %!TEX root = ../blob1.tex
 
-
 \nn{Here's the ``explicit'' version.}
 
 Fix a finite open cover of $X$, say $(U_l)_{l=1}^L$, along with an
diff -r ed2594ff5870 -r 4888269574d9 text/kw_macros.tex
--- a/text/kw_macros.tex	Fri Jun 05 16:17:31 2009 +0000
+++ b/text/kw_macros.tex	Fri Jun 05 17:41:54 2009 +0000
@@ -1,3 +1,4 @@
+%!TEX root = ../blob1.tex
 
 %%%%% excerpts from KW's include file of standard macros
 
diff -r ed2594ff5870 -r 4888269574d9 text/top_matter.tex
--- a/text/top_matter.tex	Fri Jun 05 16:17:31 2009 +0000
+++ b/text/top_matter.tex	Fri Jun 05 17:41:54 2009 +0000
@@ -1,22 +1,24 @@
-\title{Blob Homology}
-
-\author{Scott~Morrison}
-\address{
-}%
-\email{scott@tqft.net} \urladdr{http://tqft.net/}
-
-\author{Kevin~Walker}
-\address{
-}%
-\email{kevin@canyon23.net} \urladdr{http://canyon23.net/}
-
-
-\date{
-  First edition: the mysterious future
-  This edition: \today.
-}
-
-%\primaryclass{57M25} \secondaryclass{57M27; 57Q45}
-%\keywords{
-
-%}
+%!TEX root = ../blob1.tex
+
+\title{Blob Homology}
+
+\author{Scott~Morrison}
+\address{
+}%
+\email{scott@tqft.net} \urladdr{http://tqft.net/}
+
+\author{Kevin~Walker}
+\address{
+}%
+\email{kevin@canyon23.net} \urladdr{http://canyon23.net/}
+
+
+\date{
+  First edition: the mysterious future
+  This edition: \today.
+}
+
+%\primaryclass{57M25} \secondaryclass{57M27; 57Q45}
+%\keywords{
+
+%}