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85 We will use the method of acyclic models. |
85 We will use the method of acyclic models. |
86 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
86 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
87 $r$ be the restriction of $b$ to $X\setminus S$. |
87 $r$ be the restriction of $b$ to $X\setminus S$. |
88 Note that $S$ is a disjoint union of balls. |
88 Note that $S$ is a disjoint union of balls. |
89 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
89 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
90 note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
90 Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
91 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
91 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
92 so $f$ and the identity map are homotopic. |
92 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} |
93 \end{proof} |
93 \end{proof} |
94 |
94 |
95 For the next proposition we will temporarily restore $n$-manifold boundary |
95 For the next proposition we will temporarily restore $n$-manifold boundary |
96 conditions to the notation. |
96 conditions to the notation. |
97 |
97 |