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501 |
501 |
502 \medskip |
502 \medskip |
503 |
503 |
504 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
504 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
505 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
505 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
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506 (So $R_*$ is a simplicial version of the non-negative reals.) |
506 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
507 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
507 by $\iota_j$. |
508 by $\iota_j$. |
508 Define a map (homotopy equivalence) |
509 Define a map (homotopy equivalence) |
509 \[ |
510 \[ |
510 \sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
511 \sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
583 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
584 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
584 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
585 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
585 but we have come very close} |
586 but we have come very close} |
586 \nn{better: change statement of thm} |
587 \nn{better: change statement of thm} |
587 |
588 |
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589 \medskip |
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590 |
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591 Next we show that the action maps are compatible with gluing. |
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592 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
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593 the action maps $e_{X\sgl}$ and $e_X$. |
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594 The gluing map $X\sgl\to X$ induces a map |
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595 \[ |
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596 \gl: R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) \to R_*\ot CH_*(X, X) \otimes \bc_*(X) , |
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597 \] |
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598 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
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599 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. |
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600 |
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601 \medskip |
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602 |
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603 Finally we show that the action maps defined above are independent of |
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604 the choice of metric (up to iterated homotopy). |
588 |
605 |
589 \nn{...} |
606 \nn{...} |
590 |
607 |
591 |
608 |
592 \medskip\hrule\medskip\hrule\medskip |
609 \medskip\hrule\medskip\hrule\medskip |
597 \item Independence of metric, $\ep_i$, $\delta_i$: |
614 \item Independence of metric, $\ep_i$, $\delta_i$: |
598 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
615 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
599 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
616 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
600 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
617 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
601 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
618 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
602 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
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603 \end{itemize} |
619 \end{itemize} |
604 |
620 |
605 \nn{to be continued....} |
621 \nn{to be continued....} |
606 |
622 |
607 \end{proof} |
623 \end{proof} |