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123 a slightly smaller submanifold of $B$. |
123 a slightly smaller submanifold of $B$. |
124 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
124 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
125 Let $g$ be the last of the $g_j$'s. |
125 Let $g$ be the last of the $g_j$'s. |
126 Choose the sequence $\bar{f}_j$ so that |
126 Choose the sequence $\bar{f}_j$ so that |
127 $g(B)$ is contained is an open set of $\cV_1$ and |
127 $g(B)$ is contained is an open set of $\cV_1$ and |
128 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
128 $g_{j-1}(|f_j|)$ is also contained in an open set of $\cV_1$. |
129 |
129 |
130 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
130 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
131 (more specifically, $|c_{ij}| = g_{j-1}(B)$) |
131 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
132 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
132 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
133 Define |
133 Define |
134 \[ |
134 \[ |
135 s(b) = \sum_{i,j} c_{ij} + g(b) |
135 s(b) = \sum_{i,j} c_{ij} + g(b) |
136 \] |
136 \] |