0.1 CW-complex representation theorems in 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 a recent NAQAT preprint).
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.
