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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 The proof will occupy the the next several pages. |
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. |
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45 |
45 Without loss of generality, we will assume $X = Y$. |
46 Without loss of generality, we will assume $X = Y$. |
46 |
47 |
47 \medskip |
48 \medskip |
48 |
49 |
49 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
50 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
106 e_{WW'}(r\otimes b_W) = r(b_W), |
107 e_{WW'}(r\otimes b_W) = r(b_W), |
107 \] |
108 \] |
108 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
109 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
109 this case a 0-blob diagram). |
110 this case a 0-blob diagram). |
110 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
111 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
111 (by \ref{disjunion} and \ref{bcontract}). |
112 (by Properties \ref{property:disjoint-union} and \ref{property:contractibility}). |
112 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
113 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
113 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
114 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
114 such that |
115 such that |
115 \[ |
116 \[ |
116 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
117 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
151 $\supp(p)\cup\supp(b)$, and so on. |
152 $\supp(p)\cup\supp(b)$, and so on. |
152 |
153 |
153 |
154 |
154 \medskip |
155 \medskip |
155 |
156 |
156 Now for the details. |
157 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
157 |
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158 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. |
158 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. |
159 |
159 |
160 Choose a metric on $X$. |
160 Choose a metric on $X$. |
161 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
161 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
162 (e.g.\ $\ep_i = 2^{-i}$). |
162 (e.g.\ $\ep_i = 2^{-i}$). |
311 |
311 |
312 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
312 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
313 $G_*^{i,m}$. |
313 $G_*^{i,m}$. |
314 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
314 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 $\gamma_j$. |
315 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{xxxxx}. |
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}. |
317 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
317 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
318 supports. |
318 supports. |
319 Define |
319 Define |
320 \[ |
320 \[ |
321 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
321 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
608 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
608 \item prove gluing compatibility, as in statement of main thm (this is relatively easy) |
609 \item Also need to prove associativity. |
609 \item Also need to prove associativity. |
610 \end{itemize} |
610 \end{itemize} |
611 |
611 |
612 |
612 |
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613 \end{proof} |
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614 |
613 \nn{to be continued....} |
615 \nn{to be continued....} |
614 |
616 |
615 \noop{ |
617 |
616 |
618 |
617 \begin{lemma} |
619 |
618 |
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619 \end{lemma} |
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620 |
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621 \begin{proof} |
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622 |
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623 \end{proof} |
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624 |
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625 } |
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626 |
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627 |
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628 |
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629 |
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630 %\nn{say something about associativity here} |
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631 |
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632 |
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633 |
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634 |
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635 |
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