53 

    54 \begin{cor} \label{disj-union-contract}

    55 If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible.

    56 \end{cor}

    57 

    58 \begin{proof}

    59 This follows from \ref{disjunion} and \ref{bcontract}.

    60 \end{proof}

    61 

    62 Define the {\it support} of a blob diagram to be the union of all the 

    63 blobs of the diagram.

    64 Define the support of a linear combination of blob diagrams to be the union of the 

    65 supports of the constituent diagrams.

    66 For future use we prove the following lemma.

    67 

    68 \begin{lemma} \label{support-shrink}

    69 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some

    70 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$

    71 be a chain map which does not increase supports and which induces an isomorphism on

    72 $H_0(L_*)$.

    73 Then $f$ is homotopic (in $\bc_*(X)$) to the identity $L_*\to L_*$.

    74 \end{lemma}

    75 

    76 \begin{proof}

    77 We will use the method of acyclic models.

    78 Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let

    79 $r$ be the restriction of $b$ to $X\setminus S$.

    80 Note that $S$ is a disjoint union of balls.

    81 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$.

    82 note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$.

    83 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), 

    84 so $f$ and the identity map are homotopic.

    85 \end{proof}

    86