equal
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122 \[ |
122 \[ |
123 \bd(h_1(b)) = s(b) - b . |
123 \bd(h_1(b)) = s(b) - b . |
124 \] |
124 \] |
125 |
125 |
126 Next we define $h_2$. |
126 Next we define $h_2$. |
127 |
127 Let $b\in C_2$ be a 2-blob diagram. |
128 |
128 Let $B = |b|$, either a ball or a union of two balls. |
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129 By possibly working in a decomposition of $X$, we may assume that the ball(s) |
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130 of $B$ are disjointly embedded. |
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131 We will construct a 2-chain $s(b)\in \sbc_2$ such that |
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132 \[ |
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133 \bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) |
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134 \] |
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135 and the support of $s(b)$ is contained in $B$. |
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136 It then follows from \ref{disj-union-contract} that we can choose |
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137 $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
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138 |
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139 Similarly to the construction of $h_1$ above, |
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140 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
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141 of small collar maps, plus a shrunken version of $b$. |
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142 The composition of all the collar maps shrinks $B$ to a sufficiently small |
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143 disjoint union of balls. |
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144 |
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145 Let $\cV_2$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
129 |
146 |
130 \nn{...} |
147 \nn{...} |
131 |
148 |
132 |
149 |
133 |
150 |