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88 We will use the method of acyclic models. |
88 We will use the method of acyclic models. |
89 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
89 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
90 $r$ be the restriction of $b$ to $X\setminus S$. |
90 $r$ be the restriction of $b$ to $X\setminus S$. |
91 Note that $S$ is a disjoint union of balls. |
91 Note that $S$ is a disjoint union of balls. |
92 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
92 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
93 Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
93 Note that if a diagram $b'$ is part of $\bd b$ then $T(b') \sub T(b)$. |
94 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), |
94 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), |
95 so $f$ and the identity map are homotopic. |
95 so $f$ and the identity map are homotopic. |
96 \end{proof} |
96 \end{proof} |
97 |
97 |
98 For the next proposition we will temporarily restore $n$-manifold boundary |
98 For the next proposition we will temporarily restore $n$-manifold boundary |