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3 \section{The method of acyclic models} \label{sec:moam} |
3 \section{The method of acyclic models} \label{sec:moam} |
4 \todo{...} |
4 |
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5 Let $F_*$ and $G_*$ be chain complexes. |
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6 Assume $F_k$ has a basis $\{x_{kj}\}$ |
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7 (that is, $F_*$ is free and we have specified a basis). |
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8 (In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or |
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9 $k$-blob diagrams.) |
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10 For each basis element $x_{kj}$ assume we have specified a ``target" $D^{kj}_*\sub G_*$. |
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11 |
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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) |
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13 if $f(x_{kj})\in D^{kj}_*$ for all $k$ and $j$. |
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14 Let $\Compat(D^\bullet_*)$ denote the subcomplex of maps from $F_*$ to $G_*$ |
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15 such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$. |
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16 |
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17 \begin{thm}[Acyclic models] |
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18 Suppose |
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19 \begin{itemize} |
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20 \item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$ |
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21 with non-zero coefficient; |
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22 \item $D^{0j}_0$ is non-empty for all $j$; and |
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23 \item $D^{kj}_*$ is $(k{-}1)$-acyclic (i.e.\ $H_{k-1}(D^{kj}_*) = 0$) for all $k,j$ . |
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24 \end{itemize} |
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25 Then $\Compat(D^\bullet_*)$ is non-empty. |
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26 If, in addition, |
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27 \begin{itemize} |
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28 \item $D^{kj}_*$ is $m$-acyclic for $k\le m \le k+i$ and for all $k,j$, |
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29 \end{itemize} |
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30 then $\Compat(D^\bullet_*)$ is $i$-connected. |
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31 \end{thm} |
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32 |
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33 \begin{proof} |
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34 (Sketch) |
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35 This is a standard result; see, for example, \nn{need citations}. |
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36 |
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37 We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. |
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38 Choose $f(x_{0j})\in D^{0j}_0$ for all $j$. |
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39 \nn{...} |
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40 \end{proof} |