326 %(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
326 %(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
327 |
327 |
328 \begin{lemma} \label{Gim_approx} |
328 \begin{lemma} \label{Gim_approx} |
329 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
329 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
330 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
330 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
331 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
331 there exists another constant $j_{ibmn}$ such that for all $j \ge j_{ibmn}$ and all $p\in CH_n(X)$ |
332 we have $g_j(p)\ot b \in G_*^{i,m}$. |
332 we have $g_j(p)\ot b \in G_*^{i,m}$. |
333 \end{lemma} |
333 \end{lemma} |
334 |
334 |
335 For convenience we also define $k_{bmp} = k_{bmn}$ where $n=\deg(p)$. |
335 For convenience we also define $k_{bmp} = k_{bmn}$ |
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336 and $j_{ibmp} = j_{ibmn}$ where $n=\deg(p)$. |
336 Note that we may assume that |
337 Note that we may assume that |
337 \[ |
338 \[ |
338 k_{bmp} \ge k_{alq} |
339 k_{bmp} \ge k_{alq} |
339 \] |
340 \] |
340 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
341 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
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342 Additionally, we may assume that |
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343 \[ |
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344 j_{ibmp} \ge j_{ialq} |
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345 \] |
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346 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
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347 |
341 |
348 |
342 \begin{proof} |
349 \begin{proof} |
343 Let $c$ be a subset of the blobs of $b$. |
350 Let $c$ be a subset of the blobs of $b$. |
344 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
351 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
345 and all such $c$. |
352 and all such $c$. |
507 as follows. |
514 as follows. |
508 On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
515 On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
509 \[ |
516 \[ |
510 \sigma(j\ot p\ot b) = g_j(p)\ot b . |
517 \sigma(j\ot p\ot b) = g_j(p)\ot b . |
511 \] |
518 \] |
512 On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we use the track of the homotopy from |
519 On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
513 $g_j$ to $g_{j+1}$. |
520 \[ |
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521 \sigma(\iota_j\ot p\ot b) = f_j(p)\ot b , |
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522 \] |
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523 where $f_j$ is the homotopy from $g_j$ to $g_{j+1}$. |
514 |
524 |
515 Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
525 Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
516 define a version of the action map $e_X$. |
526 define a version of the action map $e_X$. |
517 A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_k$, where |
527 A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_{kbmp}$, where |
518 $k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
528 $k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
519 Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_k$. |
529 Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_{kbmp}$. |
520 The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
530 The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
521 and that $G^m_* \sup G^{m+1}_*$. |
531 and that $G^m_* \sup G^{m+1}_*$. |
522 |
532 |
523 It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
533 It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
524 to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
534 to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
527 |
537 |
528 Next we define a map |
538 Next we define a map |
529 \[ |
539 \[ |
530 e_m : G^m_* \to \bc_*(X) . |
540 e_m : G^m_* \to \bc_*(X) . |
531 \] |
541 \] |
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542 Let $p\ot b$ be a generator of $G^m_*$. |
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543 Each $g_j(p)\ot b$ or $f_j(p)\ot b$ is a linear combination of generators $q\ot c$, |
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544 where $\supp(q)\cup\supp(c)$ is contained in a disjoint union of balls satisfying |
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545 various conditions specified above. |
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546 As in the construction of the maps $e_{i,m}$ above, |
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547 it suffices to specify for each such $q\ot c$ a disjoint union of balls |
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548 $V_{qc} \sup \supp(q)\cup\supp(c)$, such that $V_{qc} \sup V_{q'c'}$ |
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549 whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
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550 |
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551 Let $q\ot c$ be a summand of $g_j(p)\ot b$, as above. |
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552 Let $i$ be maximal such that $j\ge j_{ibmp}$ |
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553 (notation as in Lemma \ref{Gim_approx}). |
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554 Then $q\ot c \in G^{i,m}_*$ and we choose $V_{qc} \sup \supp(q)\cup\supp(c)$ |
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555 such that |
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556 \[ |
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557 N_{i,d}(q\ot c) \subeq V_{qc} \subeq N_{i,d+1}(q\ot c) , |
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558 \] |
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559 where $d = \deg(q\ot c)$. |
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560 Let $\tilde q = f_j(q)$. |
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561 The summands of $f_j(p)\ot b$ have the form $\tilde q \ot c$, |
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562 where $q\ot c$ is a summand of $g_j(p)\ot b$. |
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563 Since the homotopy $f_j$ does not increase supports, we also have that |
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564 \[ |
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565 V_{qc} \sup \supp(\tilde q) \cup \supp(c) . |
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566 \] |
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567 So we define $V_{\tilde qc} = V_{qc}$. |
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568 |
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569 It is now easy to check that we have $V_{qc} \sup V_{q'c'}$ |
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570 whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
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571 As in the construction of the maps $e_{i,m}$ above, |
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572 this allows us to construct a map |
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573 \[ |
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574 e_m : G^m_* \to \bc_*(X) |
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575 \] |
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576 which is well-defined up to homotopy. |
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577 As in the proof of Lemma \ref{m_order_hty}, we can show that the map is well-defined up |
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578 to $m$-th order homotopy. |
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579 Put another way, we have specified an $m$-connected subcomplex of the complex of |
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580 all maps $G^m_* \to \bc_*(X)$. |
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581 On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
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582 One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
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583 these two maps agree up to $m$-th order homotopy. |
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584 More precisely, one can show that the subcomplex of maps containing the various |
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585 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
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586 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
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587 but we have come very close} |
532 |
588 |
533 |
589 |
534 |
590 |
535 \nn{...} |
591 \nn{...} |
536 |
592 |
541 \medskip\hrule\medskip\hrule\medskip |
597 \medskip\hrule\medskip\hrule\medskip |
542 |
598 |
543 \nn{outline of what remains to be done:} |
599 \nn{outline of what remains to be done:} |
544 |
600 |
545 \begin{itemize} |
601 \begin{itemize} |
546 \item We need to assemble the maps for the various $G^{i,m}$ into |
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547 a map for all of $CH_*\ot \bc_*$. |
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548 One idea: Think of the $g_j$ as a sort of homotopy (from $CH_*\ot \bc_*$ to itself) |
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549 parameterized by $[0,\infty)$. For each $p\ot b$ in $CH_*\ot \bc_*$ choose a sufficiently |
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550 large $j'$. Use these choices to reparameterize $g_\bullet$ so that each |
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551 $p\ot b$ gets pushed as far as the corresponding $j'$. |
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552 \item Independence of metric, $\ep_i$, $\delta_i$: |
602 \item Independence of metric, $\ep_i$, $\delta_i$: |
553 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
603 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
554 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
604 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
555 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
605 Main idea is that for all $i$ there exists sufficiently large $k$ such that |
556 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
606 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed. |
557 \item prove gluing compatibility, as in statement of main thm |
607 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
558 \item Also need to prove associativity. |
608 \item Also need to prove associativity. |
559 \end{itemize} |
609 \end{itemize} |
560 |
610 |
561 |
611 |
562 \nn{to be continued....} |
612 \nn{to be continued....} |