theorem on 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), 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 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 $(QF_j)$ , 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:

$\begin{CD} Z_j @> f_j >> QSS_j \\ @V h_{jk} VV @VV g V \\ Z_k @ > f_k >> QSS_k \end{CD}$
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.

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