
diff -r d6466180cd66 -r 3816f6ce80a8 text/evmap.tex
--- a/text/evmap.tex	Mon Apr 26 21:54:41 2010 -0700
+++ b/text/evmap.tex	Thu Apr 29 08:27:10 2010 -0700
@@ -489,12 +489,18 @@
 
 \medskip
 
+\nn{maybe wrap the following into a lemma?}
 Next we assemble the maps $e_{i,m}$, for various $i$ but fixed $m$, into a single map 
 \[
 	e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) .
 \]
 More precisely, we will specify an $m$-connected subspace of the chain complex
 of all maps from $CH_*(X, X) \otimes \bc_*(X)$ to $\bc_*(X)$.
+The basic idea is that by using Lemma \ref{Gim_approx} we can deform
+each fixed generator $p\ot b$ into some $G^{i,m}_*$, but that $i$ will depend on $b$
+so we cannot immediately apply Lemma \ref{m_order_hty}.
+To work around this we replace $CH_*(X, X)$ with a homotopy equivalent ``exploded" version
+which gives us the flexibility to patch things together.
 
 First we specify an endomorphism $\alpha$ of $CH_*(X, X) \otimes \bc_*(X)$ using acyclic models.
 Let $p\ot b$ be a generator of $CH_*(X, X) \otimes \bc_*(X)$, with $n = \deg(p)$.
@@ -514,6 +520,18 @@
 appears in the boundary of $p\ot b$ and $\alpha(p\ot b) \in G^{s,m}_*$, then
 $\alpha(q\ot c) \in G^{t,m}_*$ for some $t \le s$.
 
+If the image of $\alpha$ were contained in $G^{i,m}_*$ for fixed $i$ we could apply 
+Lemma \ref{m_order_hty} and be done.
+We will replace $CH_*(X, X)$ with a homotopy equivalent complex which affords the flexibility
+we need to patch things together.
+Let $CH^e_*(X, X)$ be the ``exploded" version of $CH_*(X, X)$, which is generated by
+tuples $(a; b_0 \sub \cdots\sub b_k)$, where $a$ and $b_j$ are simplices of $CH_*(X, X)$
+and $a\sub b_0$.
+See Figure \ref{explode_fig}.
+\nn{give boundary explicitly, or just reference hty colimit below?}
+
+\nn{this is looking too complicated; take a break then try something different}
+
 \nn{...}
 
 




