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.

   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 \]