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12 We say that a chain map $f:F_*\to G_*$ is {\it compatible} with the above data (basis and targets) |
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 $j$. |
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 $G_*$ |
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 $D^{kj}_*$. |
15 such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$. |
16 |
16 |
17 \begin{thm}[Acyclic models] |
17 \begin{thm}[Acyclic models] \label{moam-thm} |
18 Suppose |
18 Suppose |
19 \begin{itemize} |
19 \begin{itemize} |
20 \item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$ |
20 \item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$ |
21 with non-zero coefficient; |
21 with non-zero coefficient; |
22 \item $D^{0j}_0$ is non-empty for all $j$; and |
22 \item $D^{0j}_0$ is non-empty for all $j$; and |