4 \todo{...}

     4 

     5 Let $F_*$ and $G_*$ be chain complexes.

     6 Assume $F_k$ has a basis $\{x_{kj}\}$

     7 (that is, $F_*$ is free and we have specified a basis).

     8 (In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or 

     9 $k$-blob diagrams.)

    10 For each basis element $x_{kj}$ assume we have specified a ``target" $D^{kj}_*\sub G_*$.

    11 

    12 We say that a chain map $f:F_*\to G_*$ is {\it compatible} with the above data (basis and targets)

    13 if $f(x_{kj})\in D^{kj}_*$ for all $k$ and $j$.

    14 Let $\Compat(D^\bullet_*)$ denote the subcomplex of maps from $F_*$ to $G_*$

    15 such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$.

    16 

    17 \begin{thm}[Acyclic models]

    18 Suppose 

    19 \begin{itemize}

    20 \item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$

    21 with non-zero coefficient;

    22 \item $D^{0j}_0$ is non-empty for all $j$; and

    23 \item $D^{kj}_*$ is $(k{-}1)$-acyclic (i.e.\ $H_{k-1}(D^{kj}_*) = 0$) for all $k,j$ .

    24 \end{itemize}

    25 Then $\Compat(D^\bullet_*)$ is non-empty.

    26 If, in addition,

    27 \begin{itemize}

    28 \item $D^{kj}_*$ is $m$-acyclic for $k\le m \le k+i$ and for all $k,j$,

    29 \end{itemize}

    30 then $\Compat(D^\bullet_*)$ is $i$-connected.

    31 \end{thm}

    32 

    33 \begin{proof}

    34 (Sketch)

    35 This is a standard result; see, for example, \nn{need citations}.

    36 

    37 We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively.

    38 Choose $f(x_{0j})\in D^{0j}_0$ for all $j$.

    39 \nn{...}

    40 \end{proof}