# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1266992883 0
# Node ID a60332c29d0b1f44e4620132eaa51d5229df3a73
# Parent  c2d2a8f8d70ccc110fccbfc2e6b83d0a3ef23607
...

diff -r c2d2a8f8d70c -r a60332c29d0b blob1.tex
--- a/blob1.tex	Wed Feb 24 01:25:59 2010 +0000
+++ b/blob1.tex	Wed Feb 24 06:28:03 2010 +0000
@@ -21,7 +21,7 @@
 \maketitle
 
 
-[version $>$ 203;  $>$ 6 Feb 2010]
+[version $>$ 214;  $>$ 23 Feb 2010]
 
 \textbf{Draft version, read with caution.}
 
diff -r c2d2a8f8d70c -r a60332c29d0b text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Feb 24 01:25:59 2010 +0000
+++ b/text/a_inf_blob.tex	Wed Feb 24 06:28:03 2010 +0000
@@ -295,6 +295,29 @@
 It is not hard to see that this defines a chain map from 
 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
 
+Next we show that $g$ induces a surjection on homology.
+Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union 
+of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$.
+\nn{maybe should refer to elsewhere in this paper where we made a very similar argument}
+Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$.
+It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$
+retracts onto $S_*$.
+
+Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$.
+We claim that $S_{\le k}$ lies in the image of $g$.
+Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$,
+$j \le k$.
+We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls
+in $M$.
+It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is 
+disjoint from the $n{-}1$-skeleton of $K$.
+It is now easy to see that $c$ is in the image of $g$.
+
+Next we show that $g$ is injective on homology.
+
+
+
+
 \nn{...}
 
 
diff -r c2d2a8f8d70c -r a60332c29d0b text/evmap.tex
--- a/text/evmap.tex	Wed Feb 24 01:25:59 2010 +0000
+++ b/text/evmap.tex	Wed Feb 24 06:28:03 2010 +0000
@@ -113,7 +113,8 @@
 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps
 from $S$ to $T$.
 Let $\cU$ be an open cover of $S$ which affords a partition of unity.
-\nn{for some $S$ and $\cU$ there is no partition of unity?  like if $S$ is not paracompact?}
+\nn{for some $S$ and $\cU$ there is no partition of unity?  like if $S$ is not paracompact?
+in any case, in our applications $S$ will always be a manifold}
 
 \begin{lemma}  \label{extension_lemma_b}
 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$.