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1 %!TEX root = ../../blob1.tex |
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3 \section{The method of acyclic models} \label{sec:moam} |
3 \section{The method of acyclic models} \label{sec:moam} |
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5 In this section we recall the method of acyclic models for the reader's convenience. The material presented here is closely modeled on \cite[Chapter 4]{MR0210112}. |
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6 We use this method throughout the paper (c.f. Lemma \ref{support-shrink}, Theorem \ref{thm:product}, Theorem \ref{thm:gluing} and Theorem \ref{thm:map-recon}), as it provides a very convenient way to show the existence of a chain map with desired properties, even when many non-canonical choices are required in order to construct one, and further to show the up-to-homotopy uniqueness of such maps. |
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5 Let $F_*$ and $G_*$ be chain complexes. |
8 Let $F_*$ and $G_*$ be chain complexes. |
6 Assume $F_k$ has a basis $\{x_{kj}\}$ |
9 Assume $F_k$ has a basis $\{x_{kj}\}$ |
7 (that is, $F_*$ is free and we have specified a basis). |
10 (that is, $F_*$ is free and we have specified a basis). |
8 (In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or |
11 (In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or |