Talk:PlanetPhysics/Operator Algebra and Complex Representation Theorems 2
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\subsection{CW-complex representation theorems in quantum operator algebra and quantum algebraic topology}
\emph{\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} for quantum state spaces of \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} and \htmladdnormallink{quantum spin foams}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} 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 $CW$ complex 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 \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} for $i>n$ and it is a \htmladdnormallink{monomorphism}{http://planetphysics.us/encyclopedia/InjectiveMap.html} for $i=n$. The $n$-connected $CW$ model is unique up to \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} 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 \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 9 of ref. \cite{NAQAT}.
For every pair $(QSS,QSF)$ of \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} 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{\htmladdnormallink{spin networks and spin foams}{http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}.}
\begin{thebibliography}{9} \bibitem{NAQAT} I. C. Baianu, J. F. Glazebrook and R. Brown.2008.\htmladdnormallink{Non-Abelian Quantum Algebraic Topology, pp.123 Preprint}{http://planetmath.org/?op=getobj&from=papers&id=410}. \end{thebibliography}
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