330 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
330 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
331 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
331 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
332 we have $g_j(p)\ot b \in G_*^{i,m}$. |
332 we have $g_j(p)\ot b \in G_*^{i,m}$. |
333 \end{lemma} |
333 \end{lemma} |
334 |
334 |
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335 For convenience we also define $k_{bmp} = k_{bmn}$ where $n=\deg(p)$. |
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336 Note that we may assume that |
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337 \[ |
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338 k_{bmp} \ge k_{alq} |
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339 \] |
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340 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
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341 |
335 \begin{proof} |
342 \begin{proof} |
336 Let $c$ be a subset of the blobs of $b$. |
343 Let $c$ be a subset of the blobs of $b$. |
337 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
344 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
338 and all such $c$. |
345 and all such $c$. |
339 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
346 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
487 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
494 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
488 \end{proof} |
495 \end{proof} |
489 |
496 |
490 \medskip |
497 \medskip |
491 |
498 |
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 |
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528 tuples $(a; b_0 \sub \cdots\sub b_k)$, where $a$ and $b_j$ are simplices of $CH_*(X, X)$ |
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529 and $a\sub b_0$. |
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530 See Figure \ref{explode_fig}. |
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531 \nn{give boundary explicitly, or just reference hty colimit below?} |
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532 |
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533 \nn{this is looking too complicated; take a break then try something different} |
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534 |
534 |
535 \nn{...} |
535 \nn{...} |
536 |
536 |
537 |
537 |
538 |
538 |