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

Theorem 1.

Let ${[Q{F}_j]}_{j=1,\mathrm{\dots },n}$ be a complete sequence of commuting quantum spin ‘foams’ (QSFs) in an arbitrary quantum state space (QSS) (http://planetmath.org/QuantumSpaceTimes), and let $(Q{F}_j,QS{S}_j)$ be the corresponding sequence of pair subspaces of QST. If $Z_j$ is a sequence of CW-complexes such that for any $j$ , $Q{F}_j\subset Z_j$, then there exists a sequence of $n$-connected models $(Q{F}_j,Z_j)$ of $(Q{F}_j,QS{S}_j)$ and a sequence of induced isomorphisms $f_{*}{}^j:{\pi }_i(Z_j)\to {\pi }_i(QS{S}_j)$ 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 $QS{S}_j$ spaces in such a sequence. Therefore, there exists a $CW$–complex approximation of QSS defined by the sequence ${[Z_j]}_{j=1,\mathrm{\dots },n}$ of CW-complexes with dimension $n\ge 2$. This $CW$–approximation is unique up to regular homotopy equivalence.

Corollary 2.

The $n$-connected models $(Q{F}_j,Z_j)$ of $(Q{F}_j,QS{S}_j)$ form the Model Category of Quantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) $(Q{F}_j)$, whose morphisms are maps $h_{j\mathit{}k}\mathrm{:}{Z}_j\mathrm{\to }{Z}_k$ such that $h_{j\mathit{}k}\mathrm{\mid }Q{F}_j\mathrm{=}g\mathrm{:}\mathrm{(}QS{S}_j\mathrm{,}Q{F}_j\mathrm{)}\mathrm{\to }\mathrm{(}QS{S}_k\mathrm{,}Q{F}_k\mathrm{)}$, and also such that the following diagram is commutative:

$\begin{array}{ccc}\hfill Z_j\hfill & \hfill \stackrel{f_j}{\to }\hfill & \hfill QS{S}_j\hfill \\ \hfill h_{jk}\downarrow \hfill & & \hfill \downarrow g\hfill & & \\ \hfill Z_k>f_k\gg QS{S}_k\hfill \end{array}$
Furthermore, the maps $h_{j\mathit{}k}$ are unique up to the homotopy rel $Q\mathit{}{F}_j$ , and also rel $Q\mathit{}{F}_k$.

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
Canonical name	CWcomplexApproximationOfQuantumStateSpacesInQAT
Date of creation	2013-03-22 18:14:37
Last modified on	2013-03-22 18:14:37
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	29
Author	bci1 (20947)
Entry type	Theorem
Classification	msc 81T25
Classification	msc 81T05
Classification	msc 81T10
Synonym	quantum spin networks approximations by $CW$-complexes
Related topic	ApproximationTheoremForAnArbitrarySpace
Related topic	HomotopyEquivalence
Related topic	QuantumAlgebraicTopology
Related topic	ApproximationTheoremForAnArbitrarySpace
Related topic	SpinNetworksAndSpinFoams
Related topic	QuantumSpaceTimes
Defines	$CW$-complex approximation of quantum state spaces in QAT