58 but this ambiguity will not matter to us.) |
58 but this ambiguity will not matter to us.) |
59 |
59 |
60 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
60 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
61 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
61 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
62 {\it adapted to $\cU$} |
62 {\it adapted to $\cU$} |
63 \nn{or `weakly adapted'; need to decide on terminology} |
|
64 if the support of $f$ is contained in the union |
63 if the support of $f$ is contained in the union |
65 of at most $k$ of the $U_\alpha$'s. |
64 of at most $k$ of the $U_\alpha$'s. |
66 |
65 |
67 \begin{lemma} \label{extension_lemma} |
66 \begin{lemma} \label{extension_lemma} |
68 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
67 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
616 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
615 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
617 up to second order homotopy, and so on. |
616 up to second order homotopy, and so on. |
618 \end{proof} |
617 \end{proof} |
619 |
618 |
620 |
619 |
621 |
|
622 \nn{this should perhaps be a numbered remark, so we can cite it more easily} |
|
623 |
|
624 \begin{rem*} |
620 \begin{rem*} |
625 \label{rem:for-small-blobs} |
621 \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. |
622 For the proof of Lemma \ref{lem:CH-small-blobs} 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. |
623 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 |
624 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
629 of which has support close to $p(t,|b|)$ for some $t\in P$. |
625 of which has support close to $p(t,|b|)$ for some $t\in P$. |
630 More precisely, the support of the generators is contained in a small neighborhood |
626 More precisely, the support of the generators is contained in a small neighborhood |
631 of $p(t,|b|)$ union some small balls. |
627 of $p(t,|b|)$ union some small balls. |
632 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
628 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
633 \end{rem*} |
629 \end{rem*} |
634 |
|
635 |
630 |
636 |
631 |
637 \begin{prop} |
632 \begin{prop} |
638 The $CH_*(X, Y)$ actions defined above are associative. |
633 The $CH_*(X, Y)$ actions defined above are associative. |
639 That is, the following diagram commutes up to homotopy: |
634 That is, the following diagram commutes up to homotopy: |