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506 It follows that $K_{p,b}\ot b \sub K_{q,c}\ot c$ for all $q\ot c$ |
506 It follows that $K_{p,b}\ot b \sub K_{q,c}\ot c$ for all $q\ot c$ |
507 appearing in the boundary of $p\ot b$. |
507 appearing in the boundary of $p\ot b$. |
508 Thus we can apply Lemma \ref{xxxx} \nn{backward acyclic models lemma, from appendix} |
508 Thus we can apply Lemma \ref{xxxx} \nn{backward acyclic models lemma, from appendix} |
509 to get the desired map $\alpha$, well-defined up to a contractible set of choices. |
509 to get the desired map $\alpha$, well-defined up to a contractible set of choices. |
510 |
510 |
511 |
511 By construction, the image of $\alpha$ lies in the union of $G^{i,m}_*$ |
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512 (with $m$ fixed and $i$ varying). |
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513 Furthermore, if $q\ot c$ |
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514 appears in the boundary of $p\ot b$ and $\alpha(p\ot b) \in G^{s,m}_*$, then |
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515 $\alpha(q\ot c) \in G^{t,m}_*$ for some $t \le s$. |
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516 |
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517 \nn{...} |
512 |
518 |
513 |
519 |
514 |
520 |
515 |
521 |
516 |
522 |