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34 \bc_*(Y)\ar[d]^{\gl} \\ |
34 \bc_*(Y)\ar[d]^{\gl} \\ |
35 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
35 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
36 } |
36 } |
37 \end{equation*} |
37 \end{equation*} |
38 \end{enumerate} |
38 \end{enumerate} |
39 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
39 Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
40 satisfying the above two conditions. |
40 satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy. |
41 \end{thm} |
41 \end{thm} |
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42 \begin{rem} |
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43 Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this. |
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44 \end{rem} |
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45 |
42 |
46 |
43 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
47 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
44 and then give an outline of the method of proof. |
48 and then give an outline of the method of proof. |
45 |
49 |
46 Without loss of generality, we will assume $X = Y$. |
50 Without loss of generality, we will assume $X = Y$. |
343 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
347 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
344 |
348 |
345 |
349 |
346 \begin{proof} |
350 \begin{proof} |
347 |
351 |
348 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
352 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
349 (Here we are using the fact that the blobs are |
353 (Here we are using the fact that the blobs are |
350 piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
354 piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
351 We need to consider all such $c$ because all generators appearing in |
355 We need to consider all such $c$ because all generators appearing in |
352 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
356 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
353 |
357 |
580 On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
584 On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
581 One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
585 One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
582 these two maps agree up to $m$-th order homotopy. |
586 these two maps agree up to $m$-th order homotopy. |
583 More precisely, one can show that the subcomplex of maps containing the various |
587 More precisely, one can show that the subcomplex of maps containing the various |
584 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
588 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
585 \nn{now should remark that we have not, in fact, produced a contractible set of maps, |
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586 but we have come very close} |
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587 \nn{better: change statement of thm} |
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588 |
589 |
589 \medskip |
590 \medskip |
590 |
591 |
591 Next we show that the action maps are compatible with gluing. |
592 Next we show that the action maps are compatible with gluing. |
592 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
593 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |