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103 and the support of $s(b)$ is contained in $B$. |
103 and the support of $s(b)$ is contained in $B$. |
104 (If $B$ is not embedded in $X$, then we implicitly work in some stage of a decomposition |
104 (If $B$ is not embedded in $X$, then we implicitly work in some stage of a decomposition |
105 of $X$ where $B$ is embedded. |
105 of $X$ where $B$ is embedded. |
106 See Definition \ref{defn:configuration} and preceding discussion.) |
106 See Definition \ref{defn:configuration} and preceding discussion.) |
107 It then follows from Corollary \ref{disj-union-contract} that we can choose |
107 It then follows from Corollary \ref{disj-union-contract} that we can choose |
108 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
108 $h_1(b) \in \bc_2(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
109 |
109 |
110 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
110 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
111 of small collar maps, plus a shrunken version of $b$. |
111 of small collar maps, plus a shrunken version of $b$. |
112 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
112 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
113 |
113 |
129 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
129 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
130 Define |
130 Define |
131 \[ |
131 \[ |
132 s(b) = \sum_{i,j} c_{ij} + g(b) |
132 s(b) = \sum_{i,j} c_{ij} + g(b) |
133 \] |
133 \] |
134 and choose $h_1(b) \in \bc_1(X)$ such that |
134 and choose $h_1(b) \in \bc_2(X)$ such that |
135 \[ |
135 \[ |
136 \bd(h_1(b)) = s(b) - b . |
136 \bd(h_1(b)) = s(b) - b . |
137 \] |
137 \] |
138 |
138 |
139 Next we define $h_2$. |
139 Next we define $h_2$. |
250 |
250 |
251 \begin{lemma} \label{bt-contract} |
251 \begin{lemma} \label{bt-contract} |
252 $\btc_*(B^n)$ is contractible (acyclic in positive degrees). |
252 $\btc_*(B^n)$ is contractible (acyclic in positive degrees). |
253 \end{lemma} |
253 \end{lemma} |
254 \begin{proof} |
254 \begin{proof} |
255 We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_*(B^n)$. |
255 We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$. |
256 |
256 |
257 We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
257 We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
258 of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
258 of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
259 Let $r = s\circ q$. |
259 Let $r = s\circ q$. |
260 |
260 |
365 \begin{proof}[Proof of Lemma \ref{lem:bc-btc}] |
365 \begin{proof}[Proof of Lemma \ref{lem:bc-btc}] |
366 Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. |
366 Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. |
367 |
367 |
368 It suffices to show that for any finitely generated pair of subcomplexes |
368 It suffices to show that for any finitely generated pair of subcomplexes |
369 $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
369 $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
370 we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ |
370 we can find a homotopy $h:C_*\to \btc_{*+1}(X)$ such that $h(D_*) \sub \bc_{*+1}(X)$ |
371 and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. |
371 and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. |
372 |
372 |
373 By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
373 By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
374 cover $\cU$ of our choosing. |
374 cover $\cU$ of our choosing. |
375 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
375 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |