39 \nn{Also need to say something about associativity. |
39 \nn{Also need to say something about associativity. |
40 Put it in the above prop or make it a separate prop? |
40 Put it in the above prop or make it a separate prop? |
41 I lean toward the latter.} |
41 I lean toward the latter.} |
42 \medskip |
42 \medskip |
43 |
43 |
44 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, and then give an outline of the method of proof. |
44 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
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45 and then give an outline of the method of proof. |
45 |
46 |
46 Without loss of generality, we will assume $X = Y$. |
47 Without loss of generality, we will assume $X = Y$. |
47 |
48 |
48 \medskip |
49 \medskip |
49 |
50 |
50 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
51 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
51 and let $S \sub X$. |
52 and let $S \sub X$. |
52 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
53 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
53 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' |
54 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if |
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55 there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' |
54 homeomorphism $f_0 : X \to X$ so that |
56 homeomorphism $f_0 : X \to X$ so that |
55 \begin{align*} |
57 \begin{align*} |
56 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
58 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
57 \intertext{and} |
59 \intertext{and} |
58 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
60 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
311 |
313 |
312 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
314 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
313 $G_*^{i,m}$. |
315 $G_*^{i,m}$. |
314 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
316 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
315 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
317 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
316 Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{extension_lemma}. |
318 Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is |
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319 spanned by families of homeomorphisms with support compatible with $\cU_j$, |
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320 as described in Lemma \ref{extension_lemma}. |
317 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
321 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
318 supports. |
322 supports. |
319 Define |
323 Define |
320 \[ |
324 \[ |
321 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
325 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |