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%%% Primary Title: CW-complex representation theorems in QAT
%%% Primary Category Code: 02.
%%% Filename: CWComplexRepresentationTheorems.tex
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\begin{document}
\subsection{CW-complex representation theorems in quantum algebraic topology}
\emph{\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QAT.html} \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} for \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} 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 \htmladdnormallink{QSS}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} being an arbitrary, \htmladdnormallink{local quantum state space}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} (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.} (\htmladdnormallink{Baianu, Brown and Glazebrook, 2007: In Section 9 of a recent NAQAT preprint}{http://planetphysics.org/?op=getobj&from=lec&id=61}).
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}.}
\end{document}
