author | Kevin Walker <kevin@canyon23.net> |
Thu, 29 Jul 2010 19:48:59 -0400 | |
changeset 499 | 591265710e18 |
parent 492 | 833bd74143a4 |
child 500 | 5702ddb104dc |
permissions | -rw-r--r-- |
492
833bd74143a4
put in a stub appendix for MoAM, but I'm going to go do other things next
Scott Morrison <scott@tqft.net>
parents:
diff
changeset
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%!TEX root = ../../blob1.tex |
833bd74143a4
put in a stub appendix for MoAM, but I'm going to go do other things next
Scott Morrison <scott@tqft.net>
parents:
diff
changeset
|
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|
833bd74143a4
put in a stub appendix for MoAM, but I'm going to go do other things next
Scott Morrison <scott@tqft.net>
parents:
diff
changeset
|
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\section{The method of acyclic models} \label{sec:moam} |
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Let $F_*$ and $G_*$ be chain complexes. |
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Assume $F_k$ has a basis $\{x_{kj}\}$ |
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(that is, $F_*$ is free and we have specified a basis). |
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(In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or |
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$k$-blob diagrams.) |
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For each basis element $x_{kj}$ assume we have specified a ``target" $D^{kj}_*\sub G_*$. |
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We say that a chain map $f:F_*\to G_*$ is {\it compatible} with the above data (basis and targets) |
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if $f(x_{kj})\in D^{kj}_*$ for all $k$ and $j$. |
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Let $\Compat(D^\bullet_*)$ denote the subcomplex of maps from $F_*$ to $G_*$ |
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such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$. |
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\begin{thm}[Acyclic models] |
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Suppose |
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\begin{itemize} |
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\item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$ |
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with non-zero coefficient; |
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\item $D^{0j}_0$ is non-empty for all $j$; and |
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\item $D^{kj}_*$ is $(k{-}1)$-acyclic (i.e.\ $H_{k-1}(D^{kj}_*) = 0$) for all $k,j$ . |
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\end{itemize} |
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Then $\Compat(D^\bullet_*)$ is non-empty. |
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If, in addition, |
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\begin{itemize} |
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\item $D^{kj}_*$ is $m$-acyclic for $k\le m \le k+i$ and for all $k,j$, |
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\end{itemize} |
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then $\Compat(D^\bullet_*)$ is $i$-connected. |
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\end{thm} |
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\begin{proof} |
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(Sketch) |
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This is a standard result; see, for example, \nn{need citations}. |
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We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. |
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Choose $f(x_{0j})\in D^{0j}_0$ for all $j$. |
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\nn{...} |
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\end{proof} |