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 ⊂ 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 : π_i(Z_j) → π_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 ≥ 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 → Z_k such that h_jk∣QF_j = g : (QSS_j,QF_j) → (QSS_k,QF_k), and also such that the following diagram is commutative:

fj
Zj −−−→ QSSj
hjk↓ ↓g
fk
Zk −−−→ QSSk
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.

"theorem on CW--complex approximation of quantum state spaces in QAT" is owned by bci1.
(view preamble)