equal
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619 |
619 |
620 |
620 |
621 |
621 |
622 \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} |
623 |
623 |
624 \begin{rem} |
624 \begin{rem*} |
625 For the proof of xxxx below we will need the following observation on the action constructed above. |
625 \label{rem:for-small-blobs} |
|
626 For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
626 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
627 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
627 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
628 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
628 of which has support close to $p(t,|b|)$ for some $t\in P$. |
629 of which has support close to $p(t,|b|)$ for some $t\in P$. |
629 More precisely, the support of the generators is contained in a small neighborhood |
630 More precisely, the support of the generators is contained in a small neighborhood |
630 of $p(t,|b|)$ union some small balls. |
631 of $p(t,|b|)$ union some small balls. |
631 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
632 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
632 \end{rem} |
633 \end{rem*} |
633 |
634 |
634 |
635 |
635 |
636 |
636 \begin{prop} |
637 \begin{prop} |
637 The $CH_*(X, Y)$ actions defined above are associative. |
638 The $CH_*(X, Y)$ actions defined above are associative. |