OperatorAlgebraAndComplexRepresentationTheorems 2009-01-17 19:22:09 2009-01-17 19:22:09 Topic operator algebra and CW-complex representation theorems \subsection{CW-complex representation theorems in quantum operator algebra and quantum algebraic topology} \emph{QAT theorems for quantum state spaces of spin networks and quantum spin foams based on $CW$-, $n$-connected models and fundamental theorems.} Let us consider first a lemma in order to facilitate the proof of the following theorem concerning spin networks and quantum spin foams. \textbf{Lemma} \emph{Let $Z$ be a \PMlinkname{$CW$ complex}{CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams} that has the (three--dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let $f: Z \rightarrow QSS$ be a map so that $f \mid QSF = 1_{QSF}$, with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an $n$-connected $CW$ model (Z,QSF) for the pair (QSS,QSF) such that}: $f_*: \pi_i (Z) \rightarrow \pi_i (QST)$, is an isomorphism for $i>n$ and it is a monomorphism for $i=n$. The $n$-connected $CW$ model is unique up to homotopy equivalence. (The $CW$ complex, $Z$, considered here is a homotopic `hybrid' between QSF and QSS). \textbf{Theorem 2.} (\emph{Baianu, Brown and Glazebrook, 2007:}, in Section 9 of ref. \cite{NAQAT}. For every pair $(QSS,QSF)$ of topological spaces defined as in \textbf{Lemma 1}, with QSF nonempty, there exist $n$-connected $CW$ models $f: (Z, QSF) \rightarrow (QSS, QSF)$ for all $n \geq 0$. Such models can be then selected to have the property that the $CW$ complex $Z$ is obtained from QSF by attaching cells of dimension $n>2$, and therefore $(Z,QSF)$ is $n$-connected. Following \textbf{Lemma 01} one also has that the map: $f_* : \pi_i (Z) \rightarrow \pi_i (QSS)$ which is an isomorphism for $i>n$, and it is a monomorphism for $i=n$. \emph{Note} See also the definitions of (quantum) \emph{spin networks and spin foams.} \begin{thebibliography}{9} \bibitem{NAQAT} I. C. Baianu, J. F. Glazebrook and R. Brown.2008.\PMlinkexternal{Non-Abelian Quantum Algebraic Topology, pp.123 Preprint}{http://planetmath.org/?op=getobj&from=papers&id=410}. \end{thebibliography}