equal
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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. \nn{We should actually have a section with a definition of `compatible' and this statement as a lemma} |
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 |
109 \eq{ |
109 \eq{ |
110 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
110 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
111 } |
111 } |
112 The sum is over all fields $a$ on $Y$ compatible at their |
112 The sum is over all fields $a$ on $Y$ compatible at their |
113 ($n{-}2$-dimensional) boundaries with $c$. |
113 ($n{-}2$-dimensional) boundaries with $c$. |
114 `Natural' means natural with respect to the actions of diffeomorphisms. |
114 ``Natural" means natural with respect to the actions of diffeomorphisms. |
115 } |
115 } |
116 |
116 |
117 This map is very far from being an isomorphism, even on homology. |
117 This map is very far from being an isomorphism, even on homology. |
118 We fix this deficit in Section \ref{sec:gluing} below. |
118 We fix this deficit in Section \ref{sec:gluing} below. |