33 \end{enumerate} |
33 \end{enumerate} |
34 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
34 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
35 satisfying the above two conditions. |
35 satisfying the above two conditions. |
36 \end{prop} |
36 \end{prop} |
37 |
37 |
38 |
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39 \nn{Also need to say something about associativity. |
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40 Put it in the above prop or make it a separate prop? |
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41 I lean toward the latter.} |
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42 \medskip |
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43 |
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44 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
38 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
45 and then give an outline of the method of proof. |
39 and then give an outline of the method of proof. |
46 |
40 |
47 Without loss of generality, we will assume $X = Y$. |
41 Without loss of generality, we will assume $X = Y$. |
48 |
42 |
590 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
584 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
591 but we have come very close} |
585 but we have come very close} |
592 \nn{better: change statement of thm} |
586 \nn{better: change statement of thm} |
593 |
587 |
594 |
588 |
595 |
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596 \nn{...} |
589 \nn{...} |
597 |
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598 |
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599 |
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600 |
590 |
601 |
591 |
602 \medskip\hrule\medskip\hrule\medskip |
592 \medskip\hrule\medskip\hrule\medskip |
603 |
593 |
604 \nn{outline of what remains to be done:} |
594 \nn{outline of what remains to be done:} |
608 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
598 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
609 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
599 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
610 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
600 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
611 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
601 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
612 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
602 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
613 \item Also need to prove associativity. |
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614 \end{itemize} |
603 \end{itemize} |
615 |
604 |
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605 \nn{to be continued....} |
616 |
606 |
617 \end{proof} |
607 \end{proof} |
618 |
608 |
619 \nn{to be continued....} |
609 |
620 |
610 |
621 |
611 \begin{prop} |
622 |
612 The $CH_*(X, Y)$ actions defined above are associative. |
623 |
613 That is, the following diagram commutes up to homotopy: |
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614 \[ \xymatrix{ |
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615 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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616 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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617 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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618 } \] |
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619 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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620 of homeomorphisms. |
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621 \end{prop} |
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622 |
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623 \begin{proof} |
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624 The strategy of the proof is similar to that of Proposition \ref{CHprop}. |
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625 We will identify a subcomplex |
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626 \[ |
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627 G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
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628 \] |
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629 where it is easy to see that the two sides of the diagram are homotopic, then |
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630 show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
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631 |
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632 Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. |
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633 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
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634 contains $|p| \cup p\inv(|q|) \cup |b|$. |
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635 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
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636 $p(x, \cdot)\inv(|q|)$.) |
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637 |
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638 As in the proof of Proposition \ref{CHprop}, we can construct a homotopy |
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639 between the upper and lower maps restricted to $G_*$. |
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640 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
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641 that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
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642 |
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643 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
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644 to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
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645 \end{proof} |