PlanetPhysics/CW Complex Representation Theorems

CW-complex representation theorems in quantum algebraic topology

\htmladdnormallink{QAT {http://planetphysics.us/encyclopedia/QAT.html} [[../Formula/|theorems]] for [[../QuantumSpinNetworkFunctor2/|quantum state spaces]] of [[../SimplicialCWComplex/|spin networks]] and [[../TriangulationMethodsForQuantizedSpacetimes2/|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.

Lemma Let $Z$ be a $CW$ complex that has the (three--dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let $f:Z\to QSS$ be a map so that Failed to parse (syntax error): {\displaystyle f \mid QSF = 1_{QSF'' } , with [[../QuantumSpinNetworkFunctor2/|QSS]] being an arbitrary, [[../QuantumSpinNetworkFunctor2/|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}:

$f_{*}:{\pi }_i(Z)\to {\pi }_i(QST)$,

is an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] for $i>n$ and it is a [[../InjectiveMap/|monomorphism]] for $i=n$. The $n$-connected $CW$ model is unique up to [[../ThinEquivalence/|homotopy]] equivalence. (The $CW$ complex, $Z$, considered here is a homotopic `hybrid' between QSF and QSS).

Theorem 2. (Baianu, Brown and Glazebrook, 2007: In Section 9 of a recent NAQAT preprint). For every pair $(QSS,QSF)$ of [[../CoIntersections/|topological]] spaces defined as in Lemma 1 , with QSF nonempty, there exist $n$-connected $CW$ models $f:(Z,QSF)\to (QSS,QSF)$ for all $n\ge 0$. 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 Lemma 01 one also has that the map: $f_{*}:{\pi }_i(Z)\to {\pi }_i(QSS)$ which is an isomorphism for $i>n$, and it is a monomorphism for $i=n$.

Note See also the definitions of (quantum) \htmladdnormallink{spin networks and spin foams {http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}.}