81T25-CWcomplexApproximationOfQuantumStateSpacesInQAT.tex

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\pmcreated{2013-03-22 18:14:37}
\pmmodified{2013-03-22 18:14:37}
\pmowner{bci1}{20947}
\pmmodifier{bci1}{20947}
\pmtitle{$CW$-complex approximation of quantum state spaces in QAT}
\pmrecord{29}{40836}
\pmprivacy{1}
\pmauthor{bci1}{20947}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{81T25}
\pmclassification{msc}{81T05}
\pmclassification{msc}{81T10}
\pmsynonym{quantum spin networks approximations by $CW$-complexes}{CWcomplexApproximationOfQuantumStateSpacesInQAT}
%\pmkeywords{CW-complex approximation theorems in Quantum Algebraic Topology}
\pmrelated{ApproximationTheoremForAnArbitrarySpace}
\pmrelated{HomotopyEquivalence}
\pmrelated{QuantumAlgebraicTopology}
\pmrelated{ApproximationTheoremForAnArbitrarySpace}
\pmrelated{SpinNetworksAndSpinFoams}
\pmrelated{QuantumSpaceTimes}
\pmdefines{$CW$-complex approximation of quantum state spaces in QAT}

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\begin{document}
\textbf{Theorem 1.}

Let $[QF_j]_{j=1,...,n}$ be a complete sequence of commuting quantum spin `foams' 
(QSFs) in an arbitrary \PMlinkname{quantum state space (QSS)}{QuantumSpaceTimes}, and let $(QF_j,QSS_j)$ be the corresponding sequence of pair subspaces of QST. If $Z_j$ is a sequence of CW-complexes such that for any 
$j$ , $QF_j \subset Z_j$, then there exists a sequence of $n$-connected models $(QF_j,Z_j)$ of 
$(QF_j,QSS_j)$ and a sequence of induced isomorphisms  ${f_*}^j : \pi_i (Z_j)\rightarrow \pi_i (QSS_j)$ 
for $i>n$, together with a sequence of induced monomorphisms for $i=n$.
\med 

\begin{remark} 

 There exist \emph{weak} homotopy equivalences between each $Z_j$ and $QSS_j$ spaces 
in such a sequence. Therefore, there exists a $CW$--complex approximation of QSS defined by the sequence 
$[Z_j]_{j=1,...,n}$ of CW-complexes with dimension $n \geq 2$. This $CW$--approximation is 
unique up to \emph{regular} homotopy equivalence. 
\end{remark}

\textbf{Corollary 2.}

 \emph{The $n$-connected models} $(QF_j,Z_j)$ of $(QF_j,QSS_j)$ form the \emph{Model Category} of 
\PMlinkname{Quantum Spin Foams}{SpinNetworksAndSpinFoams} $(QF_j)$,  \emph{whose  morphisms are maps $h_{jk}: Z_j \rightarrow Z_k$  such that $h_{jk}\mid QF_j = g: (QSS_j, QF_j) \rightarrow (QSS_k,QF_k)$, and also such that the following diagram is commutative:} \\
 
$
\begin{CD}
Z_j @> f_j  >> QSS_j
\\ @V h_{jk} VV   @VV g V
 \\ Z_k @ > f_k >> QSS_k 
\end{CD}
$
\\
\emph{Furthermore, the maps $h_{jk}$ are unique up to the homotopy rel $QF_j$ , and also rel $QF_k$}.

\begin{remark} 
{Theorem 1} complements other data presented in the \PMlinkname{parent entry on QAT}{QuantumAlgebraicTopology}.
\end{remark}

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\end{document}