CW-complex representation theorems in quantum algebraic topology

QAT theorems for quantum state spaces of spin networks and quantum spin foams based on , -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 be a complex that has the (three–dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let be a map so that , with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an -connected model (Z,QSF) for the pair (QSS,QSF) such that:

,

is an isomorphism for and it is a monomorphism for . The -connected model is unique up to homotopy equivalence. (The complex, , 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 of topological spaces defined as in Lemma 1, with QSF nonempty, there exist -connected models for all . Such models can be then selected to have the property that the complex is obtained from QSF by attaching cells of dimension , and therefore is -connected. Following Lemma 01 one also has that the map: which is an isomorphism for , and it is a monomorphism for .

Note See also the definitions of (quantum) spin networks and spin foams.