600 |
600 |
601 \medskip |
601 \medskip |
602 |
602 |
603 Finally we show that the action maps defined above are independent of |
603 Finally we show that the action maps defined above are independent of |
604 the choice of metric (up to iterated homotopy). |
604 the choice of metric (up to iterated homotopy). |
605 |
605 The arguments are very similar to ones given above, so we only sketch them. |
606 \nn{...} |
606 Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
607 |
607 actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
608 |
608 We must show that $e$ and $e'$ are homotopic. |
609 \medskip\hrule\medskip\hrule\medskip |
609 As outlined in the discussion preceding this proof, |
610 |
610 this follows from the facts that both $e$ and $e'$ are compatible |
611 \nn{outline of what remains to be done:} |
611 with gluing and that $\bc_*(B^n)$ is contractible. |
612 |
612 As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated |
613 \begin{itemize} |
613 by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
614 \item Independence of metric, $\ep_i$, $\delta_i$: |
614 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
615 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
615 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
616 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
616 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
617 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
617 up to second order homotopy, and so on. |
618 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
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619 \end{itemize} |
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620 |
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621 \nn{to be continued....} |
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622 |
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623 \end{proof} |
618 \end{proof} |
624 |
619 |
625 |
620 |
626 |
621 |
627 \begin{prop} |
622 \begin{prop} |