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121 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
121 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
122 a slightly smaller submanifold of $B$. |
122 a slightly smaller submanifold of $B$. |
123 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
123 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
124 Let $g$ be the last of the $g_j$'s. |
124 Let $g$ be the last of the $g_j$'s. |
125 Choose the sequence $\bar{f}_j$ so that |
125 Choose the sequence $\bar{f}_j$ so that |
126 $g(B)$ is contained is an open set of $\cV_1$ and |
126 $g(B)$ is contained in an open set of $\cV_1$ and |
127 $g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$. |
127 $g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$. |
128 |
128 |
129 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
129 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
130 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
130 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
131 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
131 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
323 \BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere |
323 \BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere |
324 \end{align*} |
324 \end{align*} |
325 \end{proof} |
325 \end{proof} |
326 |
326 |
327 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
327 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
328 if there exists $a'\in \btc_k(S)$ |
328 if there exist $a'\in \btc_k(S)$ |
329 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
329 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
330 |
330 |
331 \newcommand\sbtc{\btc^{\cU}} |
331 \newcommand\sbtc{\btc^{\cU}} |
332 Let $\cU$ be an open cover of $X$. |
332 Let $\cU$ be an open cover of $X$. |
333 Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
333 Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
383 (by Lemmas \ref{bt-contract} and \ref{btc-prod}). |
383 (by Lemmas \ref{bt-contract} and \ref{btc-prod}). |
384 |
384 |
385 Now let $b$ be a generator of $C_2$. |
385 Now let $b$ be a generator of $C_2$. |
386 If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
386 If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
387 on which $b + h_1(\bd b)$ is supported. |
387 on which $b + h_1(\bd b)$ is supported. |
388 Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find |
388 Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_1(X)$, we can find |
389 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
389 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
390 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
390 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
391 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
391 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
392 |
392 |
393 The general case, $h_k$, is similar. |
393 The general case, $h_k$, is similar. |