487 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
487 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
488 \end{proof} |
488 \end{proof} |
489 |
489 |
490 \medskip |
490 \medskip |
491 |
491 |
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492 \nn{maybe wrap the following into a lemma?} |
492 Next we assemble the maps $e_{i,m}$, for various $i$ but fixed $m$, into a single map |
493 Next we assemble the maps $e_{i,m}$, for various $i$ but fixed $m$, into a single map |
493 \[ |
494 \[ |
494 e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) . |
495 e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) . |
495 \] |
496 \] |
496 More precisely, we will specify an $m$-connected subspace of the chain complex |
497 More precisely, we will specify an $m$-connected subspace of the chain complex |
497 of all maps from $CH_*(X, X) \otimes \bc_*(X)$ to $\bc_*(X)$. |
498 of all maps from $CH_*(X, X) \otimes \bc_*(X)$ to $\bc_*(X)$. |
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499 The basic idea is that by using Lemma \ref{Gim_approx} we can deform |
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500 each fixed generator $p\ot b$ into some $G^{i,m}_*$, but that $i$ will depend on $b$ |
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501 so we cannot immediately apply Lemma \ref{m_order_hty}. |
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502 To work around this we replace $CH_*(X, X)$ with a homotopy equivalent ``exploded" version |
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503 which gives us the flexibility to patch things together. |
498 |
504 |
499 First we specify an endomorphism $\alpha$ of $CH_*(X, X) \otimes \bc_*(X)$ using acyclic models. |
505 First we specify an endomorphism $\alpha$ of $CH_*(X, X) \otimes \bc_*(X)$ using acyclic models. |
500 Let $p\ot b$ be a generator of $CH_*(X, X) \otimes \bc_*(X)$, with $n = \deg(p)$. |
506 Let $p\ot b$ be a generator of $CH_*(X, X) \otimes \bc_*(X)$, with $n = \deg(p)$. |
501 Let $i = k_{bmn}$ and $j = j_i$ be as in the statement of Lemma \ref{Gim_approx}. |
507 Let $i = k_{bmn}$ and $j = j_i$ be as in the statement of Lemma \ref{Gim_approx}. |
502 Let $K_{p,b} \sub CH_*(X, X)$ be the union of the tracks of the homotopies from $g_l(p)$ to |
508 Let $K_{p,b} \sub CH_*(X, X)$ be the union of the tracks of the homotopies from $g_l(p)$ to |
511 By construction, the image of $\alpha$ lies in the union of $G^{i,m}_*$ |
517 By construction, the image of $\alpha$ lies in the union of $G^{i,m}_*$ |
512 (with $m$ fixed and $i$ varying). |
518 (with $m$ fixed and $i$ varying). |
513 Furthermore, if $q\ot c$ |
519 Furthermore, if $q\ot c$ |
514 appears in the boundary of $p\ot b$ and $\alpha(p\ot b) \in G^{s,m}_*$, then |
520 appears in the boundary of $p\ot b$ and $\alpha(p\ot b) \in G^{s,m}_*$, then |
515 $\alpha(q\ot c) \in G^{t,m}_*$ for some $t \le s$. |
521 $\alpha(q\ot c) \in G^{t,m}_*$ for some $t \le s$. |
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522 |
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523 If the image of $\alpha$ were contained in $G^{i,m}_*$ for fixed $i$ we could apply |
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524 Lemma \ref{m_order_hty} and be done. |
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525 We will replace $CH_*(X, X)$ with a homotopy equivalent complex which affords the flexibility |
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526 we need to patch things together. |
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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} |
516 |
534 |
517 \nn{...} |
535 \nn{...} |
518 |
536 |
519 |
537 |
520 |
538 |