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189 It follows from Corollary \ref{disj-union-contract} that we can choose |
189 It follows from Corollary \ref{disj-union-contract} that we can choose |
190 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
190 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
191 and with $\supp(x_k) = U$. |
191 and with $\supp(x_k) = U$. |
192 We can now take $d_j \deq \sum x_k$. |
192 We can now take $d_j \deq \sum x_k$. |
193 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
193 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
194 \nn{should have figure} |
|
195 |
194 |
196 We now define |
195 We now define |
197 \[ |
196 \[ |
198 s(b) = \sum d_j + g(b), |
197 s(b) = \sum d_j + g(b), |
199 \] |
198 \] |