$CW$-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 $QS{S}_j$ spacesin 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 isunique 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 ofQuantum 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).