485 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
485 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
486 \] |
486 \] |
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 |
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491 \medskip |
490 \medskip |
492 |
491 |
493 |
492 Next we assemble the maps $e_{i,m}$, for various $i$ but fixed $m$, into a single map |
494 \hrule\medskip\hrule\medskip |
493 \[ |
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494 e_m: CH_*(X, X) \otimes \bc_*(X) \to \bc_*(X) . |
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495 \] |
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496 More precisely, we will specify an $m$-connected subspace of the chain complex |
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497 of all maps from $CH_*(X, X) \otimes \bc_*(X)$ to $\bc_*(X)$. |
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498 |
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499 First we specify an endomorphism $\alpha$ of $CH_*(X, X) \otimes \bc_*(X)$ using acyclic models. |
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500 Let $p\ot b$ be a generator of $CH_*(X, X) \otimes \bc_*(X)$, with $n = \deg(p)$. |
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501 Let $i = k_{bmn}$ and $j = j_i$ be as in the statement of Lemma \ref{Gim_approx}. |
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502 Let $K_{p,b} \sub CH_*(X, X)$ be the union of the tracks of the homotopies from $g_l(p)$ to |
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503 $g_{l+1}(p)$, for all $l \ge j$. |
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504 This is a contractible set, and so therefore is $K_{p,b}\ot b \sub CH_*(X, X) \otimes \bc_*(X)$. |
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505 Without loss of generality we may assume that $k_{bmn} \ge k_{cm,n-1}$ for all blob diagrams $c$ appearing in $\bd b$. |
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506 It follows that $K_{p,b}\ot b \sub K_{q,c}\ot c$ for all $q\ot c$ |
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507 appearing in the boundary of $p\ot b$. |
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508 Thus we can apply Lemma \ref{xxxx} \nn{backward acyclic models lemma, from appendix} |
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509 to get the desired map $\alpha$, well-defined up to a contractible set of choices. |
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510 |
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511 |
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512 |
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513 |
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514 |
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515 |
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516 |
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517 \medskip\hrule\medskip\hrule\medskip |
495 |
518 |
496 \nn{outline of what remains to be done:} |
519 \nn{outline of what remains to be done:} |
497 |
520 |
498 \begin{itemize} |
521 \begin{itemize} |
499 \item We need to assemble the maps for the various $G^{i,m}$ into |
522 \item We need to assemble the maps for the various $G^{i,m}$ into |
505 \item Independence of metric, $\ep_i$, $\delta_i$: |
528 \item Independence of metric, $\ep_i$, $\delta_i$: |
506 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
529 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
507 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
530 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
508 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
531 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
509 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
532 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
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533 \item prove gluing compatibility, as in statement of main thm |
510 \item Also need to prove associativity. |
534 \item Also need to prove associativity. |
511 \end{itemize} |
535 \end{itemize} |
512 |
536 |
513 |
537 |
514 \nn{to be continued....} |
538 \nn{to be continued....} |