
diff -r 3816f6ce80a8 -r 5406d9423b2a text/evmap.tex
--- a/text/evmap.tex	Thu Apr 29 08:27:10 2010 -0700
+++ b/text/evmap.tex	Thu Apr 29 10:51:29 2010 -0700
@@ -332,6 +332,13 @@
 we have $g_j(p)\ot b \in G_*^{i,m}$.
 \end{lemma}
 
+For convenience we also define $k_{bmp} = k_{bmn}$ where $n=\deg(p)$.
+Note that we may assume that
+\[
+	k_{bmp} \ge k_{alq}
+\]
+for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$.
+
 \begin{proof}
 Let $c$ be a subset of the blobs of $b$.
 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ 
@@ -489,48 +496,41 @@
 
 \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 
+Let $R_*$ be the chain complex with a generating 0-chain for each non-negative
+integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$.
+Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$
+by $\iota_j$.
+Define a map (homotopy equivalence)
 \[
-	e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) .
+	\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X)
 \]
-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.
+as follows.
+On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define
+\[
+	\sigma(j\ot p\ot b) = g_j(p)\ot b .
+\]
+On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we use the track of the homotopy from 
+$g_j$ to $g_{j+1}$.
 
-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 $n = \deg(p)$.
-Let $i = k_{bmn}$ and $j = j_i$ be as in the statement of Lemma \ref{Gim_approx}.
-Let $K_{p,b} \sub CH_*(X, X)$ be the union of the tracks of the homotopies from $g_l(p)$ to 
-$g_{l+1}(p)$, for all $l \ge j$.
-This is a contractible set, and so therefore is $K_{p,b}\ot b \sub CH_*(X, X) \otimes \bc_*(X)$.
-Without loss of generality we may assume that $k_{bmn} \ge k_{cm,n-1}$ for all blob diagrams $c$ appearing in $\bd b$.
-It follows that $K_{p,b}\ot b \sub K_{q,c}\ot c$ for all $q\ot c$ 
-appearing in the boundary of $p\ot b$.
-Thus we can apply Lemma \ref{xxxx} \nn{backward acyclic models lemma, from appendix}
-to get the desired map $\alpha$, well-defined up to a contractible set of choices.
+Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually
+define a version of the action map $e_X$.
+A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_k$, where
+$k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}.
+Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_k$.
+The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex
+and that $G^m_* \sup G^{m+1}_*$.
 
-By construction, the image of $\alpha$ lies in the union of $G^{i,m}_*$ 
-(with $m$ fixed and $i$ varying).
-Furthermore, if $q\ot c$ 
-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 $t \le s$.
+It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) 
+to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$
+and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined
+up to a contractible set of choices.
 
-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?}
+Next we define a map
+\[
+	e_m : G^m_* \to \bc_*(X) .
+\]
 
-\nn{this is looking too complicated; take a break then try something different}
+
 
 \nn{...}
 




