44 \nn{should revisit above intro after this section is done}

    45 

    49 

    50 \newcommand\sbc{\bc^{\cU}}

    51 

    52 In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs)

    53 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$.

    54 

    55 \medskip

    56 

    57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted

    58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$.

    59 

    60 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is 

    61 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$.

    62 We will sometimes abuse language and talk about ``the" support of $f$,

    63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that

    64 $f$ is supported on $Y$.

    65 

    66 Fix $\cU$, an open cover of $X$.

    67 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ 

    68 of all blob diagrams in which every blob is contained in some open set of $\cU$, 

    69 and moreover each field labeling a region cut out by the blobs is splittable 

    70 into fields on smaller regions, each of which is contained in some open set of $\cU$.

    71 

    72 \begin{thm}[Small blobs] \label{thm:small-blobs-xx}

    73 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.

    74 \end{thm}

    75 

    76 \begin{proof}

    77 It suffices to show that for any finitely generated pair of subcomplexes 

    78 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$

    79 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$

    80 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$.

    81 

    82 For simplicity we will assume that all fields are splittable into small pieces, so that

    83 $\sbc_0(X) = \bc_0$.

    84 Accordingly, we define $h_0 = 0$.

    85 

    86 Let $b\in C_1$ be a 1-blob diagram.

    87 Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below.

    88 Let $B$ be the blob of $b$.

    89 

    90 

    91 \nn{...}

    92 

    93 

    94 

    95 

    96 

    97 

    98 %Let $k$ be the top dimension of $C_*$.

    99 %The construction of $h$ will involve choosing various

   100 

   101 

   102 

   103 

   104 \end{proof}

   105 

   106 

   107 

   108