616 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
616 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
617 up to second order homotopy, and so on. |
617 up to second order homotopy, and so on. |
618 \end{proof} |
618 \end{proof} |
619 |
619 |
620 |
620 |
621 \noop{ |
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622 |
621 |
623 \nn{this should perhaps be a numbered remark, so we can cite it more easily} |
622 \nn{this should perhaps be a numbered remark, so we can cite it more easily} |
624 |
623 |
625 \begin{rem} |
624 \begin{rem} |
626 For the proof of xxxx below we will need the following observation on the action constructed above. |
625 For the proof of xxxx below we will need the following observation on the action constructed above. |
627 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
626 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
628 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
627 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
629 of which has support arbitrarily close to $p(t,|b|)$ for some $t\in P$. |
628 of which has support close to $p(t,|b|)$ for some $t\in P$. |
630 This follows from the fact that the |
629 More precisely, the support of the generators is contained in a small neighborhood |
631 \nn{not correct, since there could also be small balls far from $|b|$} |
630 of $p(t,|b|)$ union some small balls. |
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631 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
632 \end{rem} |
632 \end{rem} |
633 } |
633 |
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634 |
634 |
635 |
635 \begin{prop} |
636 \begin{prop} |
636 The $CH_*(X, Y)$ actions defined above are associative. |
637 The $CH_*(X, Y)$ actions defined above are associative. |
637 That is, the following diagram commutes up to homotopy: |
638 That is, the following diagram commutes up to homotopy: |
638 \[ \xymatrix{ |
639 \[ \xymatrix{ |