0.1 CW-complex representation theorems in quantum operator algebra and quantum algebraic
topology
QAT theorems for quantum state spaces of spin networks and quantum spin foams based on CW-,
n-connected models and fundamental theorems.
Let us consider first a lemma in order to facilitate the proof of the following theorem concerning
spin networks and quantum spin foams.
Lemma Let Z be a CW complex that has the (three–dimensional) Quantum Spin ‘Foam’ (QSF) as
a subspace. Furthermore, let f : Z → QSS be a map so that f∣QSF = 1QSF , with QSS being an
arbitrary, local quantum state space (which is not necessarily finite). There exists an n-connected
CW model (Z,QSF) for the pair (QSS,QSF) such that:
f∗ : πi(Z) → πi(QST),
is an isomorphism for i > n and it is a monomorphism for i = n. The n-connected CW model is
unique up to homotopy equivalence. (The CW complex, Z, considered here is a homotopic ‘hybrid’
between QSF and QSS).
Theorem 2. (Baianu, Brown and Glazebrook, 2007:, in section 9 of ref. [1]. For every pair
(QSS,QSF) of topological spaces defined as in Lemma 1, with QSF nonempty, there exist
n-connected CW models f : (Z,QSF) → (QSS,QSF) for all n ≥ 0. Such models can be then
selected to have the property that the CW complex Z is obtained from QSF by attaching cells of
dimension n > 2, and therefore (Z,QSF) is n-connected. Following Lemma 01 one also has that
the map: f∗ : πi(Z) → πi(QSS) which is an isomorphism for i > n, and it is a monomorphism for
i = n.
Note See also the definitions of (quantum) spin networks and spin foams.
References
