# $CW$-complex approximation of quantum state spaces in QAT

Theorem 1.

Let $[QF_{j}]_{j=1,...,n}$ be a complete sequence of commuting quantum spin ‘foams’ (QSFs) in an arbitrary quantum state space (QSS) (http://planetmath.org/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 $i=n$.

###### Remark 0.1.

There exist 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 regular homotopy equivalence.

Corollary 2.

The $n$-connected models $(QF_{j},Z_{j})$ of $(QF_{j},QSS_{j})$ form the Model Category of Quantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) $(QF_{j})$, 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:

$\begin{CD}Z_{j}@>{f_{j}}>{}>QSS_{j}\\ @V{h_{jk}}V{}V@V{}V{g}V\\ Z_{k}@ >f_{k}>>QSS_{k}\end{CD}$
Furthermore, the maps $h_{jk}$ are unique up to the homotopy rel $QF_{j}$ , and also rel $QF_{k}$.

###### Remark 0.2.

Theorem 1 complements other data presented in the parent entry on QAT (http://planetmath.org/QuantumAlgebraicTopology).

Title $CW$-complex approximation of quantum state spaces in QAT CWcomplexApproximationOfQuantumStateSpacesInQAT 2013-03-22 18:14:37 2013-03-22 18:14:37 bci1 (20947) bci1 (20947) 29 bci1 (20947) Theorem msc 81T25 msc 81T05 msc 81T10 quantum spin networks approximations by $CW$-complexes ApproximationTheoremForAnArbitrarySpace HomotopyEquivalence QuantumAlgebraicTopology ApproximationTheoremForAnArbitrarySpace SpinNetworksAndSpinFoams QuantumSpaceTimes $CW$-complex approximation of quantum state spaces in QAT