492 \nn{maybe wrap the following into a lemma?}

   499 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative

   493 Next we assemble the maps $e_{i,m}$, for various $i$ but fixed $m$, into a single map 

   500 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$.

   494 \[

   501 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$

   495 	e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) .

   502 by $\iota_j$.

   496 \]

   503 Define a map (homotopy equivalence)

   497 More precisely, we will specify an $m$-connected subspace of the chain complex

   504 \[

   498 of all maps from $CH_*(X, X) \otimes \bc_*(X)$ to $\bc_*(X)$.

   505 	\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X)

   499 The basic idea is that by using Lemma \ref{Gim_approx} we can deform

   506 \]

   500 each fixed generator $p\ot b$ into some $G^{i,m}_*$, but that $i$ will depend on $b$

   507 as follows.

   501 so we cannot immediately apply Lemma \ref{m_order_hty}.

   508 On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define

   502 To work around this we replace $CH_*(X, X)$ with a homotopy equivalent ``exploded" version

   509 \[

   503 which gives us the flexibility to patch things together.

   510 	\sigma(j\ot p\ot b) = g_j(p)\ot b .

   504 

   511 \]

   505 First we specify an endomorphism $\alpha$ of $CH_*(X, X) \otimes \bc_*(X)$ using acyclic models.

   512 On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we use the track of the homotopy from 

   506 Let $p\ot b$ be a generator of $CH_*(X, X) \otimes \bc_*(X)$, with $n = \deg(p)$.

   513 $g_j$ to $g_{j+1}$.

   507 Let $i = k_{bmn}$ and $j = j_i$ be as in the statement of Lemma \ref{Gim_approx}.

   514 

   508 Let $K_{p,b} \sub CH_*(X, X)$ be the union of the tracks of the homotopies from $g_l(p)$ to 

   515 Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually

   509 $g_{l+1}(p)$, for all $l \ge j$.

   516 define a version of the action map $e_X$.

   510 This is a contractible set, and so therefore is $K_{p,b}\ot b \sub CH_*(X, X) \otimes \bc_*(X)$.

   517 A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_k$, where

   511 Without loss of generality we may assume that $k_{bmn} \ge k_{cm,n-1}$ for all blob diagrams $c$ appearing in $\bd b$.

   518 $k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}.

   512 It follows that $K_{p,b}\ot b \sub K_{q,c}\ot c$ for all $q\ot c$ 

   519 Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_k$.

   513 appearing in the boundary of $p\ot b$.

   520 The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex

   514 Thus we can apply Lemma \ref{xxxx} \nn{backward acyclic models lemma, from appendix}

   521 and that $G^m_* \sup G^{m+1}_*$.

   515 to get the desired map $\alpha$, well-defined up to a contractible set of choices.

   522 

   516 

   523 It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) 

   517 By construction, the image of $\alpha$ lies in the union of $G^{i,m}_*$ 

   524 to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$

   518 (with $m$ fixed and $i$ varying).

   525 and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined

   519 Furthermore, if $q\ot c$ 

   526 up to a contractible set of choices.

   520 appears in the boundary of $p\ot b$ and $\alpha(p\ot b) \in G^{s,m}_*$, then

   527 

   521 $\alpha(q\ot c) \in G^{t,m}_*$ for some $t \le s$.

   528 Next we define a map

   522 

   529 \[

   523 If the image of $\alpha$ were contained in $G^{i,m}_*$ for fixed $i$ we could apply 

   530 	e_m : G^m_* \to \bc_*(X) .

   524 Lemma \ref{m_order_hty} and be done.

   531 \]

   525 We will replace $CH_*(X, X)$ with a homotopy equivalent complex which affords the flexibility

   532 

   526 we need to patch things together.

   533 

   527 Let $CH^e_*(X, X)$ be the ``exploded" version of $CH_*(X, X)$, which is generated by

   528 tuples $(a; b_0 \sub \cdots\sub b_k)$, where $a$ and $b_j$ are simplices of $CH_*(X, X)$

   529 and $a\sub b_0$.

   530 See Figure \ref{explode_fig}.

   531 \nn{give boundary explicitly, or just reference hty colimit below?}

   532 

   533 \nn{this is looking too complicated; take a break then try something different}