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85 $r$ be the restriction of $b$ to $X\setminus S$. |
85 $r$ be the restriction of $b$ to $X\setminus S$. |
86 Note that $S$ is a disjoint union of balls. |
86 Note that $S$ is a disjoint union of balls. |
87 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
87 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
88 Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
88 Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
89 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
89 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
90 so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma} |
90 so $f$ and the identity map are homotopic. \nn{We should actually have a section \S \ref{sec:moam} with a definition of ``compatible" and this statement as a lemma} |
91 \end{proof} |
91 \end{proof} |
92 |
92 |
93 For the next proposition we will temporarily restore $n$-manifold boundary |
93 For the next proposition we will temporarily restore $n$-manifold boundary |
94 conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$. |
94 conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$. |
95 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
95 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |