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.

   129 By possibly working in a decomposition of $X$, we may assume that the ball(s)

   130 of $B$ are disjointly embedded.

   131 We will construct a 2-chain $s(b)\in \sbc_2$ such that

   132 \[

   133 	\bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b)

   134 \]

   135 and the support of $s(b)$ is contained in $B$.

   136 It then follows from \ref{disj-union-contract} that we can choose

   137 $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$.

   138 

   139 Similarly to the construction of $h_1$ above, 

   140 $s(b)$ consists of a series of 2-blob diagrams implementing a series

   141 of small collar maps, plus a shrunken version of $b$.

   142 The composition of all the collar maps shrinks $B$ to a sufficiently small 

   143 disjoint union of balls.

   144 

   145 Let $\cV_2$ be an auxiliary open cover of $X$, satisfying conditions specified below.